GMAT coordinate geometry note: What was the (arithmetic mean) temperature reading of the light bulb? In a certain dance troupe, there are 55 women and 33 men.


GMAT coordinate geometry note: If all of the women are 62 inches tall and all of the men are 70 inches tall, what is the average height of the dancers in the troupe?


GMAT word problem note: Track mean, median, range, distribution, and how operations change spread. Anton and Lena start at the same point on a circular track, measuring 10 meters in diameter, and begin walking counterclockwise at the same time, with Anton walking more quickly than Lena.


GMAT word problem note: When Anton has traveled exactly halfway around the track, they both stop walking. They then observe that the distance between them along a straight line measures exactly meters, as shown above.


GMAT word problem note: What fraction of the track has Lena covered? Eddy, Mario, and Peter have $32, $72, and $98, respectively.


GMAT word problem note: They pool their money and redistribute the entire amount among themselves. If Eddy now has the median amount of money in the group, what is the greatest amount of money that Eddy could now have?


GMAT coordinate geometry note: The portion of the coordinate plane that lies within the square, but above the line, is then shaded as shown above. If A is the area of the shaded region in square units, which of the following specifies all the possible values of A?


GMAT geometry note: Because the formula for the area of a rectangle is A = lw, rephrase the question. The area of a quadrilateral is maximized when the quadrilateral is a square, so first try l = w = 6.


GMAT geometry note: In this case, the area is 36 cm2 and the answer to the question is Yes. If, on the other hand, l = 11.5 and w = 0.5, the perimeter is still 24, but the area is (11.5)(0.5) = 5.75 cm2.


GMAT geometry note: In this case, the answer to the question is No. The diagonal of the rectangle is less than 11.


GMAT geometry note: If the diagonal is less than 11, then the sides must also be less than 11. If l = w = 6, then the diagonal is shorter than 11 and, as last time, the area is 36.


GMAT geometry note: In this case, the answer to the question is Yes. If, on the other hand, l = 3 and w = 1, the diagonal is less than 11, but the area is (3)(1) = 3 cm2.


GMAT geometry note: The sides must be less than 11 and the perimeter must be 24. The case of the square still maximizes the area: l = w = 6 and the area is 36.


GMAT geometry note: The largest possible length is just under 11, making the width just over 1. (<11) (>1) = something larger than 10.


GMAT geometry note: The area must be greater than 10.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. The problem describes an ant walking along a number line in a complex way.


GMAT word problem note: It starts at 0, then walks to the right until it reaches a multiple of Then, it walks to the le until it reaches a multiple of 2. The problem specifies that the ant must walk to the le at least one unit.


GMAT word problem note: Therefore, even if it is already standing on a multiple of 2, it must walk to the next one. In this case, it walks le until it reaches 4:.


GMAT word problem note: Then it walks to the right again until it reaches a new multiple of 5:. The question asks how far the ant walks, in total, to reach the number 50 for the first time.


GMAT word problem note: Since this is too many units to realistically count by hand, there must be an underlying pattern. Write down the first few distances that the ant walks, using your number line.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. There is a repeating cycle of four numbers.


GMAT statistics note: Starting at 5 on the number line, the ant moves le 1, then right 6, then le 2, then right 7, which takes it to 15 on the number line. That is, the ant walks a total of 1 + 6 + 2 + 7 = 16 units to move from 5 to 15.


GMAT statistics note: Then it walks another 16 units to move from 15 to 25, from 25 to 35, and from 35 to 45. In total, the ant walks 4(16) = 64 units to move from 5 to 45.


GMAT statistics note: To get from 45 to 50, the ant first walks one unit le to 44, then six units right to 50. In total, the ant walks 64 + 1 + 6 = 71 units to move from 5 to 50.


GMAT statistics note: Finally, add the five units it walks from the beginning: 71 + 5 = 76. It is also possible to estimate the answer.


GMAT statistics note: As established above, traveling 10 points on the number line (from 15 to 25, from 25 to 35, and so on) requires the ant to travel 16 units. To begin at 0 and end at 50, the ant would travel approximately 5(16) = 80 units.


GMAT statistics note: The problem indicates that a, b, and c are positive and asks whether a is greater than , which is the average (or arithmetic mean) of b and c. Draw a picture and rephrase the question: “On the number line, is a positioned to the right of the m.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. It isn’t clear whether b or c is the larger value—the point on the right—so a could be closer to the smaller number (making a less than the average) or the larger number (making a greater than the average).


GMAT word problem note: This statement indicates nothing about a, so it can’t be sufficient. Together, the two statements indicate that b is the point on the right, so a must be on the right side of the midpoint.


GMAT word problem note: The length of any side of the cube is 4, and the problem asks for the distance between the center of the cube and any of its vertices (corners). Chop up the cube into eight smaller cubes to see that the distance from the center of the 4 × 4 × 4 cube to any corner is the diagonal of a 2 × 2 × 2 cube.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. You can find the diagonal of a cube in a variety of ways.


GMAT statistics note: Probably the fastest (besides applying a memorized formula) is to use the “Super Pythagorean” theorem, which extends to three dimensions:. In the special case when the three sides of the box are equal, as they are in a cube, use this equation:.


GMAT statistics note: To visualize this set of weighted averages problems, imagine a teeter-totter that is 50 meters long, marked off from 0 to 50 to represent scores. One hundred people of equal weight sit on the teeter-totter at their respective scores.


GMAT statistics note: The weighted average is the position where the teeter-totter would balance. Thus, to answer whether the average score is greater than 30, take extremes according to the given conditions and see whether you can swing the balance to either side of 30 (or whether you are forced to balance on one side of 30 only).


GMAT quant note: Track mean, median, range, distribution, and how operations change spread. More than 70 people scored 40 or higher.


GMAT quant note: If all 100 scored 40 (or higher), then the average is above 30. If, on the other hand, 71 people scored 40 and the other 29 scored 0, does the average drop to 30 or lower?


GMAT quant note: Match up the 29 people who scored 0 with 29 of the people who scored 40. These 58 people together have an average score of 20.


GMAT quant note: The other 42 people have a score of 40. If you average these two groups, the average must be below 30, since 58 is larger than 42.


GMAT quant note: Seventy-five people scored 40 or higher. If all 100 scored 40, then the average is higher than 30.


GMAT quant note: If 75 scored 40 and the other 25 scored 0, then what? Match the 25 people who scored 0 with 25 people who scored 40.


GMAT quant note: These 50 people have an average score of 20. The remaining 50 people have an average score of 40, so the overall average is exactly 30.


GMAT quant note: It is possible, therefore, to have an average that is higher than 30 or an average that is not higher than 30.


GMAT number properties note: Track mean, median, range, distribution, and how operations change spread. Fewer than 10 people scored 50.


GMAT number properties note: Say that 1 person scored 50 and the other 99 scored 0. The average is definitely below 30.


GMAT number properties note: If on the other hand, 1 person scored 50 and the other 99 scored 40, then the average is definitely above 30. Each score was achieved by no more than 2 people.


GMAT number properties note: There are 51 integers between 0 and 50, inclusive. In other words, there are 102 possible scores to spread among the test-takers.


GMAT number properties note: Since there are 100 test-takers, almost every score is taken. The highest possible average will occur when nobody scores 0 points.


GMAT number properties note: If 2 test- takers score 1, 2 test-takers score 2, and so on up to 50, then the average score will be approximately 25. As this is the highest allowable average for this scenario, it’s impossible for the average to be greater than 30.


GMAT number properties note: The average looks like it’s in the 112° range. Calculate the over/under with 112° as the assumed baseline.


GMAT number properties note: The four 110° readings are each 2° below the baseline, and the three 111° readings are each 1° below the baseline, for a total of 11° below baseline. On the other side, the 113°, 114°, and 115° readings are a total of 11° above the baseline.


GMAT number properties note: The average is exactly 112°.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Because Anton has walked exactly halfway around the track, the distance between Anton and the starting point is the diameter.


GMAT word problem note: Since is approximately 5(1.7) = 8.5, the line between Anton and Lena is only slightly shorter than the diameter of the circle.


GMAT geometry note: To calculate the exact answer, observe that Anton, Lena, and the starting line form the three vertices of a triangle. Because one side of the triangle is the diameter of the circle, the triangle is a right triangle with hypotenuse 10.


GMAT geometry note: The known leg is equal to You can either use the Pythagorean theorem to calculate the missing leg (the straight line between Lena and the starting point) or you can note that the known leg and the hypotenuse have a ratio of , so the triangle must be a 30–60–90 triangle and the remaining side must equal 5.


GMAT geometry note: The arc of the circle that Lena traveled has a corresponding inscribed angle of 30 degrees (the vertex at point A). The corresponding central angle is twice the inscribed angle, or 60 degrees.


GMAT geometry note: Alternatively, the distance between Lena and the starting line is equal to the radius of the circle. Therefore, the triangle with vertices at Lena, the starting line, and the center of the circle is an equilateral triangle, and all of its angles equal 60 degrees.


GMAT coordinate geometry note: Track mean, median, range, distribution, and how operations change spread. The pool of money is $32 + $72 + $98 = $202.


GMAT coordinate geometry note: A er the redistribution, each person will have an amount between $0 and $202, inclusive. Call the amounts L, M, and H (low, median, high).


GMAT coordinate geometry note: To maximize M (Eddy’s share), minimize L and H. Minimum L = $0 Minimum H = M Maximum M = Total pool of money − Minimum L − Minimum H M = $202 − $0 − M 2M = $202 M = $101 10.


GMAT coordinate geometry note: On your paper, draw the square in the coordinate plane. The line passes through the point (0, 4), so label that point as well.


GMAT coordinate geometry note: Since the slope of the line could vary, sketch several possible lines that pass through the point, being certain to test extreme cases. If the slope is very positive, then the line is nearly vertical, and the area above the line will be very close to 2(8) = 16.


GMAT geometry note: As the slope decreases, the area above the line will decrease. If the slope is 1, the area above the line is the area of a triangle with base 4 and height 4, which is equal to Then, as the slope continues to decrease, the area above the line will once again increase steadily.


GMAT geometry note: The greatest area is found when the slope is very negative, since the area will be slightly less than (6)(8) = 48.


GMAT geometry note: The area of the shaded region can be a minimum of exactly 8 and can be a maximum of almost, but not equal to, 48 (since the line cannot be vertical).


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Hybrid Problems Identify and Sequence the Parts Where to Start Minor Hybrids.


GMAT statistics note: Percent problems usually relate part, whole, rate, and percent change. Try these four hybrid problems.


GMAT statistics note: Try-It #8-1 Set A consists of four consecutive positive integers. Set B is constructed as follows: each integer in set A is randomly either increased by 10% or decreased by 10%, and the four resulting values compose set B.


GMAT statistics note: Try-It #8-2 Set S contains 100 consecutive integers. If the range of the negative elements of set S equals 80, what is the average (arithmetic mean) of the positive numbers in the set?


GMAT statistics note: The smallest integer in set A was increased by 10% when set B was constructed. The greatest integer in set A was increased by 10% when set B was constructed.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Try-It #8-3 If a and b are consecutive positive integers, is ab divisible by 30?


GMAT sets, probability, and counting note: Try-It #8-4 A carnival card game gives the contestant a one in three probability of successfully choosing the correct card and thereby winning the game. If a contestant plays the game repeatedly, what is the minimum number of times that he must play the game so that the probability that he never loses is less than 1%?


GMAT sets, probability, and counting note: A2 is divisible by 25.(1) 63 is a factor of b2.(2).


GMAT geometry note: Hybrid Problems Hybrid problems blend topics together. They contain two or more qualitatively different kinds of obstacles that you must surmount on the way to the answer.


GMAT geometry note: Some hybrid problems feature content areas that are fairly closely related. Other hybrid problems feature content areas that share little in common.


GMAT geometry note: These problems must be solved in separate steps. The difficulty of a hybrid problem is related to the following questions:.


GMAT geometry note: Problems with minor additional content areas are generally easier to solve than hybrids that blend topics together in an unusual, fundamental, and How closely related are the subjects being tested? Are the content areas covered in the same section of the Manhattan Prep All the Quant guide?


GMAT geometry note: The more closely related the subjects, the easier it will be to navigate the problem. How important is each of the subject areas?


GMAT geometry note: Are each of the subject areas fundamental to the problem or is one of them just a low-level disguise that can be quickly disposed of? The more important each topic is in solving the problem, the more difficult the problem will usually be.


GMAT quant note: Focus on equations, expressions, variables, and step-by-step manipulation. The best hybrids are one of a kind.


GMAT quant note: You will have to bring your A game to solve them. That’s why you’re reading this chapter!


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Identify and Sequence the Parts When you encounter a hybrid problem, first pick out all of the topics tested.


GMAT number properties note: Take a look at the first problem from the pop quiz:. Try-It #8-1 Set A consists of four consecutive positive integers.


GMAT number properties note: Set B is constructed as follows: each integer in set A is randomly either increased by 10% or decreased by 10%, and the four resulting values compose set B. This problem refers to consecutive integers, but it involves percents and also asks for the range of a set of numbers, which is a statistics concept.


GMAT number properties note: Which aspect of the problem should you tackle first? In this problem, a set begins as four consecutive integers and is then changed.


GMAT number properties note: So you might start your approach by jotting down some sets of consecutive integers—either as numbers or in terms of a variable. Then, The greatest integer in set A was increased by 10% when set B was constructed.


GMAT number properties note: The smallest integer in set A was increased by 10% when set B was constructed.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Apply the percent change described in the problem and, finally, check the range of the resulting set.


GMAT fractions, decimals, and percents note: In general, start with wherever you feel the logical beginning is or whichever part you feel is an easier, cleaner starting point.


GMAT number properties note: Percent problems usually relate part, whole, rate, and percent change. Where to Start STARTING AT THE BEGINNING As you contemplate the logical order of steps, you might feel less confident with the second stage than with the first.


GMAT number properties note: If so, go ahead and start at the beginning. Just articulate very clearly, “What intermediate result will I get once I’m finished with the first part of the problem?” Begin by defining set A.


GMAT number properties note: Set A consists of four consecutive integers, such as 1, 2, 3, 4, or 10, 11, 12, 13. Next, choose a statement to work with first.


GMAT number properties note: The other integers might have been either increased or decreased. Using one of your cases, calculate what might happen to the set:.


GMAT number properties note: Set A Scenarios Set B 1, 2, 3, 4 All values increased by 10% 1.1, 2.2, 3.3, 4.4 1, 2, 3, 4 Smallest value increased by 10%, other values decreased by 10% 1.1, 1.8, 2.7, 3.6 Finally, calculate the range of the resulting set. The range of a set is the difference between the greatest number and the smallest number in the set.


GMAT statistics note: Percent problems usually relate part, whole, rate, and percent change. Set A Scenarios Set B Range All values increased by 10% 1.1, 2.2, 3.3, Smallest value increased by 10%, other values decreased by 10% 1.1, 1.8, 2.7, Since different ranges are possible, the statement is insufficient.


GMAT statistics note: STARTING AT THE END You might decide that it is easier to start at the end and work backwards. Just ask yourself, “What information do I need to have as a last step before arriving at a solution to the question?” In this problem, you might first consider what would cause set B to have a different range.


GMAT statistics note: The range of a set is the difference between the greatest and smallest numbers in the set. If the difference between these two numbers can vary, the range can vary as well.


GMAT statistics note: However, the smallest integer might have increased by 10%, bringing it closer to the greatest integer, or it might have decreased by 10%, bringing it farther from the greatest integer.


GMAT statistics note: Percent problems usually relate part, whole, rate, and percent change. Now, put the two statements together.


GMAT statistics note: One option is to try a few different cases, making sure to increase the smallest integer in set A and decrease the greatest integer in set A. Set A Scenario Set B Range 1, 2, 3, 4 All values increased by 10% 1.1, 2.2, 3.3, 4.4 4.4—1.1 = 3.3 Try another case, using different numbers:.


GMAT statistics note: Set A Scenario Set B Range 10, 11, 12, 13 All values increased by 10% 11, 12.1, 13.2, 14.3 14.3—11 = 3.3 The range is the same, but this doesn’t mean that the statements are sufficient together. To see why, use a different approach.


GMAT statistics note: Set A always contains the numbers x, x + 1, x + 2, and x + 3. When the smallest value in set A is increased by 10%, it goes from x to 1.1x.


GMAT statistics note: When the greatest value is increased by 10%, it goes from x + 3 to 1.1x + 3.3. So these two values differ by 3.3.


GMAT statistics note: However, you can’t assume that the smallest and greatest values in set A always become the smallest and greatest values in set B. If two of the values switch places, the range might change.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Set A Scenario Set B Range Smallest and largest values increased by 10%, other values decreased by 10% The range of the set might vary, even when both statements are used.


GMAT statistics note: Try-It #8-2 Set S contains 100 consecutive integers. If the range of the negative elements of set S equals 80, what is the average (arithmetic mean) of the positive numbers in the set?


GMAT statistics note: In this problem, information is given about the range of numbers in a set. Thus, knowing how to work with statistics techniques will be important in solving the problem.


GMAT statistics note: Solving this problem also requires using consecutive integers techniques— namely, counting consecutive integers and computing their average. These two topics are closely related but still cover different ideas.


GMAT statistics note: If you start with the statistics piece of the problem, then you’ll be able to find the highest and lowest negative numbers in the set of consecutive integers. This will act as an input to the formulas for computing the largest and smallest positive integers in the set, and subsequently the average of the positive integers in the set.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. First, determine what set of consecutive negative integers will result in a range of 80.


GMAT number properties note: Range is defined as the difference between the highest and lowest numbers in a set:. The high number among the negative terms is the largest negative integer, Therefore, the lowest number in set S is −81.


GMAT number properties note: Use this result to jump to the consecutive integers portion of the solution. In this case, −81 is the smallest or “first” element in the set:.


GMAT number properties note: Therefore, the highest or “last” number in set S is 18. Finally, calculate the average of the positive terms in the set using 1 as the smallest positive integer (“first pos”) and 18 as the largest positive integer (“last pos”).


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The average of the positive numbers in the set is 9.5.


GMAT number properties note: If instead you decided to start with the second step, you would write the formula for the average of consecutive integers first, focusing on the positive integers:. The two unknowns you need to solve for are the greatest positive integer and the least positive integer because the question asks only about the positive integers in the set.


GMAT number properties note: Some of the integers in set S are negative and some are positive, so clearly the least positive integer will be 1. Therefore, you only need to figure out what the greatest integer in the set will be.


GMAT number properties note: This is the information needed in the last step of solving the problem. In order to calculate this number, you would now need to apply the definition of range:.


GMAT number properties note: The greatest negative integer (the “high” number) is −1.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Therefore, the least number in the set is −81.


GMAT number properties note: Plug this into the formula for counting consecutive integers:. Therefore, the highest number in the set is 18.


GMAT number properties note: Finally, plug 18 into the average formula:. Whether you start solving this problem from the beginning or from the end, study how the steps hook together: the output of one step becomes the input to another.


GMAT number properties note: These are the “turns” you have to make in solving any hybrid problem. O en, a number that plays one role in a particular formula or calculation plays a completely different role in the next step.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Minor Hybrids In minor hybrid problems, one of the following conditions applies:.


GMAT number properties note: These problems can be easier to solve than major hybrids. However, you will still benefit greatly from paying close attention to the turns as you move through the stages of the solution.


GMAT number properties note: Try-It #8-3 If a and b are consecutive positive integers, is ab divisible by 30? This problem tests your skill with both Divisibility & Primes and Consecutive Integers.


GMAT number properties note: Consecutive integer concepts o en lend themselves well to questions about divisibility. The content areas in the problem are closely related.


GMAT number properties note: For instance, they are covered in the same section of the Manhattan Prep All the Quant guide. One of the content areas is a low-level disguise or some other minor feature.


GMAT number properties note: A2 is divisible by 25.(1) 63 is a factor of b2.(2).


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. To be divisible by 30, a number must have 2, 3, and 5 as prime factors.


GMAT number properties note: Thus, the question becomes: “Does ab have 2, 3, and 5 as prime factors?” Next, use a concept from consecutive integers: a and b are consecutive positive integers. Thus, either a or b is an even number, which means that the product ab is automatically divisible by 2.


GMAT number properties note: The question can be further simplified: “Does the product ab have 3 and 5 as prime factors?” (1) INSUFFICIENT:. Knowing that 5 is a prime factor of a indicates that 5 is a factor of ab, but does not indicate whether 3 is a factor.


GMAT number properties note: Knowing that 3 is a prime factor of b indicates that 3 is a factor of ab, but does not indicate whether 5 is a factor. According to the combined statements, a is divisible by 5 and b is divisible by 3.


GMAT number properties note: This is sufficient information to answer the rephrased question. Notice that the solution never indicated exactly what consecutive integers a and b are—this isn’t necessary.


GMAT number properties note: Also notice a potential trap in this problem—assuming that a < b bec.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Try-It #8-4 A carnival card game gives the contestant a one in three probability of successfully choosing the correct card and thereby winning the game.


GMAT sets, probability, and counting note: If a contestant plays the game repeatedly, what is the minimum number of times that he must play the game so that the probability that he never loses is less than 1%? This problem primarily tests Probability theory.


GMAT sets, probability, and counting note: In addition, Exponents are needed to represent the impact of playing multiple games on the probability of the outcomes. The probabilities are given in fractions, yet the question is asked in terms of percents, so Fraction, Decimal, and Percent (FDP) connections are relevant.


GMAT sets, probability, and counting note: Finally, the question is phrased in terms of an inequality, so Inequalities come into play. “The probability that he never loses” can be rephrased as “the probability that he always wins.” This probability can be expressed as , where is the chance of winning on a single play and n is the number of times the contestant plays.


GMAT sets, probability, and counting note: Number of Plays P(All Wins) Approx.


GMAT sets, probability, and counting note: Percent problems usually relate part, whole, rate, and percent change. Number of Plays P(All Wins) Approx.


GMAT sets, probability, and counting note: Equivalent You might also have noticed that only the denominator in the probabilities mattered, as the numerator was always 1. To have a probability of less than 1%, the fractional probability must be “1 over something greater than 100.” In order for 3n >100, n must be at least 5.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Problem Set Solve the following problems and identify the topics being tested.


GMAT number properties note: The average (arithmetic mean) of a list of six numbers is equal to 0. What is the positive difference between the number of positive numbers in the list and the number of negative numbers in the list?


GMAT number properties note: Each of the positive numbers in the list equals 10.(1) Each of the negative numbers in the list equals −5.(2) 2. If a, b, and c are positive, is a > b?


GMAT number properties note: If c is randomly chosen from the integers 20 to 99, inclusive, what is the probability that c3 − c is divisible by 12?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If x and y are positive integers greater than 1 such that x − y and are both even integers, which of the following numbers must be non-prime integers?


GMAT number properties note: A number cube with faces numbered 1 through 6 has an equal chance of landing on any face when rolled. If the number cube is rolled twice, what is the probability that the sum of the two rolls is a prime number?


GMAT number properties note: The function {x} is defined as the area of a square with diagonal of length x. If x > 0 and {x2} = x2, what is the value of x?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. (1 centimeter = 10 millimeters) 9.


GMAT number properties note: Three consecutive integers are selected from the integers 1 to 50, inclusive. What is the sum of the remainders that result when each of the three integers is divided by x?


GMAT number properties note: When the greatest of the consecutive integers is divided by x, the remainder is 0. When the least of the consecutive integers is divided by x, the remainder is 1.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If x, y, and z are all distinct positive integers, and the percent increase from x to y is equal to the percent increase from y to z, what is x?


GMAT quant note: Track mean, median, range, distribution, and how operations change spread. Since the average of the six numbers in the list is 0, the sum of the six numbers is 0.


GMAT quant note: There could be positive numbers and negative numbers in the set. Zero is not mentioned, but this does not rule it out.


GMAT quant note: In order for the sum of the numbers in the set to be 0, either all the terms are 0 or there are some positives and some negatives. The list could be {−2, −2, −2, −2, −2, 10}, and the positive difference between the number of positive terms and the number of negative terms would be 4.


GMAT quant note: Alternatively, the list could be {−20, −20, 10, 10, 10, 10}, and the positive difference would be 2. The set could be {−5, 1, 1, 1, 1, 1}, and the positive difference between the number of positive terms and the number of negative terms would be 4.


GMAT quant note: The set could be {−5, −5, −5, 5, 5, 5}, and the positive difference would be 0. The statements together suggest that the set has twice as many −5 terms as 10 terms, in order to maintain a sum of 0.


GMAT quant note: If every term is negative or positive, then the set would have.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Terms are possible, so the set could be {−5, −5, 0, 0, 0, 10}, and the positive difference would be 1.


GMAT quant note: You could simplify the numerator arithmetically (multiply out the terms and then add). Alternatively, factor a 22 out of the numerator.


GMAT quant note: Then, distribute the denominator (which becomes the difference of squares). If a, b, and c are all positive, then a > b + positive.


GMAT quant note: Therefore, a must be greater than b. You could also prove this fact by Testing Cases.


GMAT quant note: Case b + c < a Is this case possible according to (2)? A > b b + positive < number greater than b Possible a = b b + c < b Impossible, since c is positive a < b b + c < number less than b Impossible, since c is positive.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Only the a > b case is possible, so the answer is a definite Yes.


GMAT algebra note: A(a + c) > b(b + c) a2 + ac > b2 + bc a2 + ac − b2 − bc > 0 a2 − b2 + ac − bc > 0 Group similar terms to simplify. [a2 − b2] + [ac − bc] > 0 Note the Quadratic Template.


GMAT algebra note: (a − b)(a + b) + c(a − b) > 0 (a − b)[(a + b) + c] > 0 Factor out a − b. Therefore, a + b + c is positive, because all three additive terms are positive.


GMAT algebra note: By Number Properties sign rules, (a − b) must also be positive in order for the product to be greater than 0. It is hard to see where to begin or what series of manipulations will be productive.


GMAT algebra note: LT = less than and GT = greater than. GTb means “a number greater than b.” Case Is this case possible according to (1)?


GMAT algebra note: The le is greater than the right.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Case Is this case possible according to (1)?


GMAT number properties note: The two expressions are equal. The le side is actually less than the right side, not greater than as (1) requires.


GMAT number properties note: The words divisible and probability are used, so this question is about Divisibility & Primes and Probability. There are 99 − 20 + 1 = 80 possible values for c, so the unknown is how many of these c values yield a c3 − c that is divisible by 12.


GMAT number properties note: The prime factorization of 12 is 2 × 2 × 3. There are at least two ways to think about this: numbers are divisible by 12 if they are divisible by 3 and by 2 twice, or if they are multiples of both 4 and 3.


GMAT number properties note: The expression involving c can be factored. These are consecutive integers.


GMAT number properties note: It may help to put them in increasing order: (c − 1)c(c + 1). Thus, this question has a lot to do with Consecutive Integers, and not only because the integers 20 to 99 themselves are consecutive.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. In any set of three consecutive integers, a multiple of 3 will be included.


GMAT number properties note: Thus, (c − 1)c(c + 1) is always divisible by 3 for any integer c. This takes care of part of the 12.


GMAT number properties note: So the question becomes: “How many of the possible (c − 1)c(c + 1) values are divisible by 4?” Since the prime factors of 4 are 2’s, it makes sense to think in terms of Odds and Evens. Thus, c3 − c is divisible by 12 whenever c is odd, possibilities.


GMAT number properties note: Thus, c3 − c is also divisible by 12 whenever c is a multiple of 4, which are the cases c = possibilities. X and y are both positive and is an integer, so x > y.


GMAT number properties note: Since x − y is even, either x and y are both even or they are both odd. Since = an even integer, x = y × (even integer).


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Therefore, x is an even integer, as is y.


GMAT number properties note: X and y are both positive even integers and x > y. No even number greater than 2 is prime, so x can’t be prime.


GMAT number properties note: X and y are each positive even integers and x > y. Thus, x + y is even and the least possible value of x + y = 4 + 2 = 6.


GMAT number properties note: All even numbers greater than or equal to 6 are non-prime. It could be that x = 4 and y = 2, so , which is not prime, but is also not an integer.


GMAT number properties note: In fact, if an even integer, then III. First, think about the prime numbers less than 12, the maximum sum of the numbers.


GMAT number properties note: These primes are 2, 3, 5, 7, 11. The probability of rolling 2, 3, 5, 7, or 11 is equal to the number of ways to roll any of these sums divided by the total number of possible outcomes.


GMAT number properties note: The total number of possible outcomes is 6 × 6 = 36. Sum of 2 can happen 1 way: 1 + 1 Sum of 3 can happen 2 ways: 1 + 2, 2 + 1 Sum of 5 can happen 4 ways: 1 + 4, 2 + 3, 3 + 2, 4 + 1 Sum of 7 can happen 6 ways: 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2, 6 + 1.


GMAT coordinate geometry note: Sum of 11 can happen 2 ways: 5 + 6, 6 + 5 That’s a total of 1 + 2 + 4 + 6 + 2 = 15 ways to roll a prime sum. The problem defines the function in words.


GMAT coordinate geometry note: You’ll need to translate into math. “The function {x} is defined as the area of a square with diagonal of length x.” {x} = s2, where s is the side of the square.


GMAT coordinate geometry note: If the diagonal equals x, then the side of the square equals The question stem also indicates that {x2} = x2. In other words, applying the defined function to x2 will result in an answer of x2.


GMAT coordinate geometry note: Now, set that equal to x2 and solve. It is acceptable to divide by x2 because the question stem indicates that x ≠ 0.) Therefore, x is equal to or .


GMAT coordinate geometry note: Since the question stem indicates that x > 0, only is a valid solution. Alternatively, you could work backwards from the answers.


GMAT coordinate geometry note: Next, the question specifies that that becomes {2}, so what does this function return? “The function {x} is defined as the area of a square with diagonal of length x.” The function {2} is defined as the area of a square with diagonal length 2.


GMAT coordinate geometry note: The side length of this square is , so the area of this square is . Therefore, {2} = 2, which matches the question stem specification that {x2} = x2.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Microchip radius = (2.5 cm)(10 mm/cm) = 25 mm 9.


GMAT number properties note: When consecutive integers are divided by the same number, the remainders follow a repeating pattern. For instance, when consecutive integers are divided by 4, the remainders form a repeating pattern of [1, 2, 3, 0], with every fourth integer being divisible by 4.


GMAT number properties note: This statement indicates that the greatest of the three integers is divisible by x. However, the sum of the remainders depends on the value of x.


GMAT number properties note: For instance, if x = 10, the three remainders would be 8, 9, and 0, respectively, and their sum would be 17. If x = 5, the three remainders would be 3, 4, and 0, with a sum of 7.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. However, the sum of the remainders again depends on the value of x.


GMAT number properties note: If x = 10, the three remainders would be 1, 2, and 3, with a sum of 6. If x = 2, the three remainders would be 1, 0, and 1, with a sum of 2.


GMAT number properties note: Call the three consecutive integers y, y + 1, and y + 2. The remainder when y is divided by x is 1 and the remainder when y + 2 is divided by x is 0.


GMAT number properties note: Write out the known remainders and write out enough other terms to see where the pattern repeats. In this case, the term y – 1 also has a remainder of 0, so the pattern repeats every three terms.


GMAT number properties note: Y − 1 y y + 1 y + 2 The only way for the remainders to have a consistent repeating pattern is for the missing remainder to equal 2 (in which case x must equal 3). Therefore, the sum of the three remainders is 1 + 2 + 0 = 3.


GMAT number properties note: The question asks for the value of x. Translate the equation given in the question stem.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The problem stem specifies that x is an integer, so the right-hand side of the equation must also be an integer.


GMAT number properties note: Therefore, z must be a factor of y2. The question asks for the value of x.


GMAT number properties note: If y is prime, then there are two possible scenarios. This isn’t allowed, though, because the question stem indicates that the variables represent three different positive integers.


GMAT number properties note: Case 2: y2 = z, in which case x must equal 1. If z = 9, then y2 is a multiple of 9, so z must be a multiple of 3.


GMAT number properties note: It’s possible that z = 9, y = 3, and x = 1. It’s also possible that z = 9, y = 36, and x = 4.


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. Workout Set 1 Workout Set 2 Workout Set 3 Workout Set 4 Workout Set 5 Workout Set 6 Workout Set 7 Workout Set 8 Workout Set 9 Workout Set 10 Workout Set 11 Workout Set 12 Workout Set 13 Workout Set 14 Workout Set 15 Workout Set 16.


GMAT geometry note: The circle with center O has a circumference of . If AC is a diameter of the circle, what is the length of BC?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. A batch of widgets costs p + 15 dollars for a company to produce and each batch sells for p(9 − p) dollars.


GMAT number properties note: For which of the following values of p, does the company make a profit? If K is the sum of the reciprocals of the consecutive integers from 41 to 60 inclusive, which of the following is less than K?


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Triplets Adam, Bruce, and Charlie enter a triathlon.


GMAT sets, probability, and counting note: There are nine competitors in the triathlon. If every competitor has an equal chance of winning, and three medals will be awarded, what is the probability that at least two of the triplets will win a medal?


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Floyd started at one end of a trail at 8:07 a.m.


GMAT word problem note: And hiked the entire 5 miles to the other end of the trail at an average rate of 3 miles per hour. Connie started at the same end of the trail at 8:30 a.m.


GMAT word problem note: And hiked the entire length of the trail at an average rate of 2.5 miles per hour. Finally, Karen started her hike in the same place at 10:00 a.m.


GMAT word problem note: And hiked the entire length of the trail at an average rate of 4 miles per hour. During approximately what percent of the 4-hour period from 8:00 a.m.


GMAT word problem note: To noon were at least two of the three hikers on the trail?


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Half an hour a er Car A started traveling from Newtown to Oldtown, a distance of 62 miles, Car B started traveling along the same road from Oldtown to Newtown.


GMAT word problem note: The cars met each other on the road 15 minutes a er Car B started its trip. If Car A traveled at a constant rate that was 8 miles per hour greater than Car B’s constant rate, how many miles had Car B driven when they met?


GMAT word problem note: If x = 2b − (88 + 86), for which of the following b values is x closest to zero?


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. If y is 20% less than 90% of x and z is 25% more than 130% of y, then z is what percent of x?


GMAT fractions, decimals, and percents note: Set A consists of three consecutive positive multiples of 3, and set B consists of five consecutive positive multiples of 5.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Integers in set A is equal to the sum of the integers in set B, what is the least number that could be a member of set A?


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.


GMAT geometry note: If AC is a diameter of the circle, then triangle ABC is a right triangle, with angle ABC = 90 degrees. The shortest side of a triangle is across from its least angle and the longest side of a triangle is across from its greatest angle.


GMAT geometry note: The circumference of the circle = , so . Thus, AC ≈ 10.2 and BC < 10.2.


GMAT geometry note: The value should be less than 9. Since angle ACB is 30 degrees, angle CAB is 60 degrees.


GMAT geometry note: The sides in a 30–60–90 triangle have the ratio , so use the ratio to compute the desired side, BC.


GMAT strategy note: Watch divisibility, factors, remainders, parity, and prime structure. You can work backwards from the answers.


GMAT strategy note: The company makes a profit when p = 4. Alternatively, profit equals revenue minus cost.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. For p > 5, both (p − 5) and (p − 3) are positive.


GMAT fractions, decimals, and percents note: In that case, the profit is negative (i.e., the company loses money). The profit is only positive if (p − 5) and (p − 3) have opposite signs, which occurs when 3 < p < 5.


GMAT fractions, decimals, and percents note: The sum has 20 fractional terms. It is impossible to compute this by hand in two minutes.


GMAT fractions, decimals, and percents note: Instead, look at the maximum and minimum possible values for the sum. The greatest fraction in the sum is .


GMAT fractions, decimals, and percents note: K is definitely less than 20 × , or , which is less than Minimum:. The least fraction in the sum is .


GMAT fractions, decimals, and percents note: K is definitely greater than 20 × Therefore, < K < I.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. With nine competitors and only three medals awarded, only of the competitors will win overall.


GMAT sets, probability, and counting note: Although a simplification, it is reasonable for each competitor to see his or her chance of winning a medal as , or to expect to win of a medal (pretending for a moment that medals can be shared). The question asks for the probability that at least two of the triplets will win a medal.


GMAT sets, probability, and counting note: In other words, you want to of the triplets to win medals or for each triplet to win to of a medal. Since and are both greater than , you are looking for the probability that the triplets will win medals at a rate greater than that expected for competitors overall.


GMAT sets, probability, and counting note: In other words, this would be an unusual outcome. Thus, the probability should be less than .


GMAT sets, probability, and counting note: You could then guess from among the remaining answers with a 1 in 3 chance of guessing correctly. To solve, use the probability formula and combinatorics.


GMAT quant note: Separate total outcomes from desired outcomes and count carefully. First, find the total number of outcomes for the triathlon.


GMAT quant note: Three will win medals and six will not. Set up an anagram grid where Y represents a medal, N no medal.


GMAT quant note: Competitor C1 C2 C3 C4 C5 C6 C7 C8 C9 Medal Y Y Y N N N N N N Therefore, the number of ways three medals can be awarded is Now, determine the number of instances when at least two brothers win a medal. Practically speaking, this could happen when 1) exactly three brothers win or 2) exactly two brothers win.


GMAT quant note: Start with all three triplets winning medals, where Y represents a medal. Triplet A B C Non-triplet C1 C2 C3 C4 C5 C6 Medal Y Y Y Medal N N N N N N.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. The number of ways this could happen is .


GMAT sets, probability, and counting note: This makes sense, as there is only one instance in which all three triplets would win medals and all of the other competitors would not. (If you recognize this immediately, no need to write out the math.) Next, calculate the instances when exactly two of the triplets win medals.


GMAT sets, probability, and counting note: Triplet A B C Non-triplet C1 C2 C3 C4 C5 C6 Medal Y Y N Medal Y N N N N N Since both triplets and non-triplets win medals in this scenario, consider the possibilities for both sides of the grid.


GMAT sets, probability, and counting note: For the triplets, the number of ways that two could win medals is For the non-triplet competitors, the number of ways that one could win the remaining medal is Multiply these two numbers to get the total number of instances: 3 × 6 = The brothers win at least two medals in 18 + 1 = 19 cases. The total number of cases is 84, so the probability is.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. To answer the question, you’ll need to identify the periods of time during which different numbers of hikers are on the trail.


GMAT word problem note: Next, identify the time period during which each hiker was on the trail. Floyd traveled the 5 miles at a rate of 3 miles per hour (mph), so he spent = 1 hour and 40 minutes on the trail.


GMAT word problem note: Therefore, Floyd was on the trail from 8:07 to 9:47. Connie traveled the 5 miles at a rate of 2.5 mph, so she spent = 2 hours on the trail.


GMAT word problem note: Connie was on the trail from 8:30 to Finally, Karen traveled the 5 miles at a rate of 4 mph, so she spent = 1 hour 15 minutes on the trail. Karen was on the trail from 10:00 to 11:15.


GMAT word problem note: Plot each of these time periods on your drawing.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. There were no periods of time during which all three hikers were on the trail.


GMAT word problem note: However, there were two periods during which exactly two of the three hikers were on the trail. Floyd and Connie were both on the trail from 8:30 to 9:47, and Connie and Karen were both on the trail from 10:00 to 10:30.


GMAT word problem note: This represents a total of 1 hour and 47 minutes, or approximately 1.75 hours. 75 hours is more than a third of 4 hours, but less than half of 4 hours.


GMAT word problem note: It is also possible to simplify the fraction more precisely.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Note that you must convert 15 minutes to 0.25 hours.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Rate Time Distance Car A (r + 8) mph 0.75 hours (0.75) (r + 8) miles Car B r mph 0.25 hours 0.25r miles Total 62 miles Set up and solve an equation for the total distance.


GMAT word problem note: Therefore, in 15 minutes, Car B traveled a distance of 0.25r = (0.25)(56) = 14 miles. Testing the choices would be a natural way to solve this problem, since the question doesn’t ask you to solve for b in general, but rather “for which of the following is x closest to zero?” However, numbers between 220 and 242 are too great to plug and compute.


GMAT word problem note: Instead, manipulate the terms with base 8 to see how they might balance with 2b.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. If k = 2, then which is less than 1.


GMAT algebra note: Stop if you can tell that an answer won’t equal the target. Alternatively, solve the problem algebraically.


GMAT algebra note: The expression given is of the form , where and.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. First, try to eliminate the radical signs entirely, leaving only a2 and b2 in the denominator.


GMAT algebra note: To do so, multiply by a fraction that is a convenient form of 1. Notice the “difference of two squares” special product created in the denominator.


GMAT algebra note: An algebraic setup would be fairly ugly on this problem, but a combination of two other techniques work quite nicely: choose smart numbers and estimate. There are no real values for the variables, so Choosing Smart Numbers is a valid approach.


GMAT algebra note: Percent problems usually relate part, whole, rate, and percent change. That you can estimate if the Smart Numbers get a little messy.


GMAT algebra note: Of the three variables, it is easiest to pick for x. If you were to pick for y or z, you would have to do “reverse” calculations to find the other variables.


GMAT algebra note: Since this is a percent problem, try 100 first. E.g., Since x = 100, 90% of x is equal to 90.


GMAT algebra note: Therefore, you is equal to 20% less than 90 = 90 − 18 = 72 This number is a little ugly, so round down to 70. Therefore, y = 70, so 130% of y is equal to 70 + 21 = 91.


GMAT algebra note: The next step requires you to take 25% of that number, so 91 is going to get messy. Estimate again, but this time round up a little bit (to offset the error you introduced when you rounded down earlier).


GMAT algebra note: Round to the nearest number that is divisible by 4, which is 92. Next, find 25% more than 92.


GMAT algebra note: Z = 92 + 23 = 115 If x = 100 and z ≈ 115, then the final calculation is which is closest to 117%. The values in set A and set B are unknown.


GMAT algebra note: Jot them down in terms of variables. If x is a positive integer, then x, x + 1, and x + 2 are consecutive integers, and 3x, 3(x + 1), and 3(x + 2) are three consecutive.


GMAT number properties note: Track mean, median, range, distribution, and how operations change spread. Similarly, set B contains 5y, 5(y + 1), 5(y + 2), 5(y + 3), and 5(y + 4), where y is a positive integer.


GMAT number properties note: Do not use the same variable for both sets, since it is unclear whether or how they are related. Sum the numbers in each set.


GMAT number properties note: Set A: 3x + 3(x + 1) + 3(x + 2) = 3(3x + 3) Set B: 5y + 5(y + 1) + 5(y + 2) + 5(y + 3) + 5(y + 4) = 5(5y + 10) The sums of the sets are equal. Set them equal to each other and simplify as much as possible.


GMAT number properties note: 3(3x + 3) = 5(5y + 10) 9(x + 1) = 25(y + 2) Because x and y are positive integers, the right side of the equation is a multiple of 25 and the le side is a multiple of 9. In other words, each side of the equation must be a multiple of both 9 and 25.


GMAT number properties note: Because 9 and 25 have no common factors, x + 1 must be a multiple of 25 and y + 2 must be a multiple of 9. The least possible value for x is x = 24, and the least number that could be a member of set A is 3x = 3(24) = 72.


GMAT number properties note: Manually add them up or note that for an evenly spaced set the sum is (# of terms) × (the median term). The integers in set B are multiples of 5, so the sum of set B is a multiple of 5.


GMAT number properties note: By extension, the (equivalent) sum of set A must be a multiple of 5.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. If m = 4n + 9, where n is a positive integer, what is the greatest common factor of m and n?


GMAT quant note: M = 9s, where s is a positive integer(1) n = 4t, where t is a positive integer(2) 13. A museum sold 30 tickets on Saturday.


GMAT quant note: Some of the tickets sold were $10 general exhibit tickets and the rest were $70 special exhibit tickets. How many general exhibit tickets did the museum sell on Saturday?


GMAT quant note: The museum’s total revenue from ticket sales on Saturday was greater than $1,570 and less than $1,670. The museum sold more than 20, but fewer than 25, special exhibit tickets on Saturday.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If x and y are integers, is xy = yx?


GMAT number properties note: What is the perimeter of isosceles triangle ABC? The length of side AB is 9.(1) The length of side BC is 4.(2) 17.


GMAT number properties note: If x, y, and z are integers and 2x 5y z = 6.4 × 106, what is the value of xy?


GMAT number properties note: Track mean, median, range, distribution, and how operations change spread. A school has a students and b teachers.


GMAT number properties note: If a < 150, b < 25, and classes have a maximum of 15 students, can the a students be distributed among the b teachers so that each class has the same number of students? (Assume that any student can be taught by any teacher.) It is possible to divide the students evenly into groups of 2, 3, 5, 6, 9, 10, or 15.


GMAT number properties note: The greatest common factor of a and b is 10.(2) 20. A set of five distinct positive integers has a median of 3 and a range of 12.


GMAT number properties note: What is the mean of the set of integers? The product of the integers in the set is a multiple of 14.(1) The sum of the integers in the set is a multiple of 13.(2).


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. You can cross-multiply the inequality, as long as you consider both cases.


GMAT quant note: This is a conditional rephrased question—importantly, one with completely opposite questions as possibilities. Any statement that doesn’t at least answer the question of whether a − b is positive or negative is unlikely to be sufficient.


GMAT quant note: You would also have to carry each statement through both questions, so this path is not very efficient. It’s better to use the original inequality to test cases in order to eliminate the incorrect answers.


GMAT quant note: Test cases to determine whether this statement is sufficient. A b Is Case 1 2 1 ✓ No Case 2 2 Yes If a − b is negative, the question becomes “Is 1 < ab(a − b)?” (FLIPPED inequality sign) If a − b is positive, the question becomes “Is 1 > ab(a − b)?” (ORIGINAL inequality sign).


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. A b a < b Is Case 1 Yes Case 2 ?


GMAT quant note: No (1) AND (2) INSUFFICIENT:. Whenever possible, reuse cases you’ve already tested (you can only do this when the case makes both statements true).


GMAT quant note: No Once again, two valid sets of numbers return two different answers, so even together, the statements are insufficient.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The greatest common factor (GCF) of any two numbers is given by the product of the shared or overlapping primes in the two numbers.


GMAT number properties note: Any such question can be rephrased accordingly: “What exactly are the overlapping factors of m and n?” (1) SUFFICIENT:. This statement indicates that m must be a multiple of 9.


GMAT number properties note: If m is a multiple of 9 and m = 4n + 9, then this equation is really saying. Multiple of 9 = 4n + 9 In order for this to be true, 4n must also be a multiple of 9.


GMAT number properties note: The number 4 does not contain any factors of 9, so n itself must be a multiple of 9. If both m and n are multiples of 9, and m is exactly 9 units away from 4n, then the greatest possible common factor is 9, so this statement is sufficient.


GMAT number properties note: (If you’re not sure about the logic of that last part, test out a couple of real numbers. n must be a multiple of 9.


GMAT number properties note: If n = 9, then m = 4(9) + 9 = 45. The GCF of the numbers 9 and 45 is 9.


GMAT number properties note: If n = 18, then m = 4(18) + 9 = 81. The GCF of the numbers 18 and 81 is still 9.) Alternatively, methodically list possible values of s, m, and n.


GMAT number properties note: Ignore any case for which the statement is not true. For the remaining cases, determine the answer to the question.


GMAT number properties note: S = pos int m = 9s n = pos int GCF of m and n?


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. S = pos int m = 9s n = pos int GCF of m and n?


GMAT fractions, decimals, and percents note: Zero ✗ Fraction ✗ Fraction ✗ Fraction ✗ Fraction ✗ Fraction ✗ Fraction ✗ Fraction ✗.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. In this case, n is only a positive integer when (s − 1) is a multiple of 4.


GMAT sequences and patterns note: Also, m and n are both multiples of 9. Try the next number in the pattern, 9 + 4 = 13, to be confident that the apparent GCF pattern will continue.


GMAT sequences and patterns note: S = pos int m = 9s n = pos int GCF of m and n? 5 (9)(5) = 45 9 (9)(9) = 81 13 (9)(13) = 117 The variables m and n always share an overlapping factor of 9, but there is never any overlap between their remaining factors.


GMAT sequences and patterns note: The GCF is always (2) INSUFFICIENT: n = 4t, where t is a positive integer. In this case, n must be a multiple of 4.


GMAT statistics note: Watch divisibility, factors, remainders, parity, and prime structure. T = pos int n = 4t m = 4n+ 9 GCF of m and n?


GMAT statistics note: 1 4 16 + 9 = 25 1 2 8 32 + 9 = 41 1 3 12 48 + 9 = 57 3 There is more than one possible GCF, so the statement is not sufficient. Use the integer constraint to test possible cases.


GMAT statistics note: Because the special tickets are so much more expensive, begin by choosing numbers for which (70)(# of special tickets) is approximately $1,600. If g = 7, total revenue = 10(7) + 70(23) = 70 + 1,610 = 1,680.


GMAT statistics note: If g = 8, total revenue = 10(8) + 70(22) = 80 + 1,540 = 1,620. If g = 9, total revenue = 10(9) + 70(21) = 90 + 1,470 = 1,560.


GMAT statistics note: Only g = 8 gives a total revenue in the given range. Don’t assume that having a range of values automatically means a statement is insufficient to answer a Value question.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Constraint (such as an integer constraint) may limit the number of cases to just one possibility.


GMAT algebra note: Alternatively, you can use algebra, though that path is a bit long on this problem. Then (30 − g) represents the number of special exhibit tickets sold.


GMAT algebra note: Set up and solve the following inequality:. The only integer between 7.2 and 8.8 is 8, so g must be 8.


GMAT algebra note: The museum sold 8 general exhibit tickets. This statement indicates that the museum sold 21, 22, 23, or 24 special exhibit tickets.


GMAT algebra note: Since the museum sold a total of 30 tickets, this means that it sold 9, 8, 7, or 6 general exhibit tickets. Test cases on this theory problem.


GMAT algebra note: If you can generate both Yes and No cases, you will prove insufficiency.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Case 1 2 0 20 = 1 02 = 0 No Case 2 3 1 31 = 3 13 = 1 No Case 3 4 2 42 = 16 24 = 16 Yes (2) INSUFFICIENT:.


GMAT number properties note: Case 1 8 1 81 = 8 18 = 1 No Case 2 4 2 42 = 16 24 = 16 Yes (1) AND (2) SUFFICIENT:. Test whether any cases work for both statements.


GMAT number properties note: The case x = 4 and y = 2 was already used for both statements and returned a Yes answer. Can any of the No cases work for both statements?


GMAT number properties note: If you are good with numbers, you can try to find another pair of integers that will work with both equations. Since x and y have to be integers, there are only a limited number of possibilities.


GMAT number properties note: The two numbers need to multiply to 4 and.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. In addition to 4 and 2, the pair −4 and −2 fit the bill, as long as x = −2 and y = −4.


GMAT algebra note: Alternatively, solve algebraically to determine the possible values for x and y. You have already tested the case where x = 4 (and y = 2), so test the other case.


GMAT algebra note: Yes In either case, xy = yx. Odd exponents do not “hide the sign” of the base.


GMAT algebra note: If ab3c4 > 0, then a and b must have the same sign so that their product is positive.


GMAT geometry note: Case, a3b must also be positive. As a result, in order for the inequality a3bc5 to be positive, c5 must be positive.


GMAT geometry note: The rephrased question is, “Is c (1) INSUFFICIENT:. No information about the sign of c.


GMAT geometry note: Answers the rephrased question directly. The perimeter of a triangle is equal to the sum of the three sides.


GMAT geometry note: Knowing the length of one side of the triangle is not enough to find the sum of all three sides. Triangle ABC is an isosceles triangle, which means that two of the sides are equal in length.


GMAT geometry note: The statements provide two of the side lengths, so the third side, AC, must equal one of the given sides. There is a hidden constraint in this problem: the triangle must be valid.


GMAT geometry note: Recall that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. AB BC AC Perimeter Valid triangle?


GMAT geometry note: 9 4 4 9 + 4 + 4 = 17 No: 4 + 4 < 9.


GMAT geometry note: AB BC AC Perimeter Valid triangle? 9 4 9 9 + 4 + 9 = 22 Yes: 4 + 9 > 9 A “triangle” with three sides of 4, 4, and 9 is not really a triangle, as it cannot be drawn with those dimensions.


GMAT geometry note: Therefore, the actual sides of the triangle must be AB = 9, BC = 4, and AC = 9. Express both sides of the equation in terms of prime numbers.


GMAT geometry note: The right side of the equation is composed of only 2’s and 5’s. The le side of the equation has x number of 2’s and y number of 5’s along with some factor z.


GMAT geometry note: This unknown factor z must be composed of only 2’s and/or 5’s, or it must be 1 (i.e., with no prime factors). If z = 1, then x = 11 and y = 5.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If z = 2?5?, where the exponents are not 0, then x and y will depend on the value of those exponents.


GMAT number properties note: The rephrased question is thus, “How many factors of 2 and 5 are in z?” (1) SUFFICIENT:. If z = 20 = 2251, then this answers the rephrased question.


GMAT number properties note: Incidentally, this implies that 2x5y(2251) = 21155, so x = 9 and y = 4, making xy = 36 (though you do not have to solve for x and y). (2) INSUFFICIENT: x = 9, but the statement doesn’t indicate anything about the value of y.


GMAT number properties note: This problem deals with both squares and inequalities. negative numbers become positive when squared and that squaring a number less than 1 results in a value that is even closer to 0.


GMAT number properties note: Without any more information about z, this statement only tells you that x2 is greater than y2. However, x could be either greater than y or less than y, depending on whether x is positive or negative.


GMAT number properties note: Since all squares are positive, x2 is much greater than y2. In this case, x2 is greater than y2 and x is greater than y.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. This statement does not provide any information about y, so y could be greater than, less than, or equal to x.


GMAT word problem note: This implies that |x| is greater than |z|. There are only two situations in which x is greater than z and the absolute value of x is greater than the absolute value of z.


GMAT word problem note: One possibility is that x is positive and z is a lesser positive number. Another possibility is that x is positive and z is a negative number with an absolute value that is less than x.


GMAT word problem note: These scenarios are illustrated below. Therefore, x is definitely positive.


GMAT word problem note: Since x is positive, |x| = x, so x > |y|. If x is greater than the absolute value of y, x must also be greater than y itself, so the answer to the question is definitely Yes.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. In order for the a students to be distributed evenly among the b teachers, a must be divisible by b.


GMAT number properties note: This is a factor question in disguise: “Is a divisible by b?” (1) INSUFFICIENT:. The given numbers represent factors of a.


GMAT number properties note: If the a students can be divided evenly into a group of 2, then 2 is a factor of a. Likewise, 3 and 5 must be factors of a.


GMAT number properties note: If 2 and 3 are already factors of a, then 6 must also be a factor (since 2 × 3 = 6). Likewise, you’ve already counted one factor of 3, so you need only one more factor of 3 to make 9.


GMAT number properties note: You already have the necessary factors to make both 10 and 15, so ignore those numbers as well. The final list of factors is: 2, 3, 5, and 3.


GMAT number properties note: As a result, a must equal (2)(3)(5)(3) = 90 or a multiple of 90. Since the question stem indicated that a < 150, a must actually be 90.


GMAT number properties note: This statement provides no information about the teachers, however, so it is insufficient. If the greatest common factor (GCF) of a and b is 10, then a and b must both be multiples of 10.


GMAT number properties note: Since the question stem indicated that b < 25, b must be 10 or 20. The value of a is any multiple of 10 up to 140, inclusive.


GMAT number properties note: This is not enough to determine whether a is divisible by b.


GMAT number properties note: Track mean, median, range, distribution, and how operations change spread. All five of the integers in the set are positive, and all of the integers are distinct, or different from each other.


GMAT number properties note: Jot down blanks to represent the values in the set. The least two integers in the set must be less than the median, different from each other, and positive: the first two numbers must be 1 and 2.


GMAT number properties note: Finally, the range of the set, or the difference between the greatest and least integers, is 12. Therefore, the greatest integer in the set is 13.


GMAT number properties note: The only unknown value in the set is the second-greatest integer, which must be between 3 and 13, exclusive. Finding this integer will give you enough information to find the mean of the set.


GMAT number properties note: For a number to be a multiple of 14, it must be a multiple of both 2 and 7. The product of the four known values is already a multiple of 2, since the set already contains the number 2.


GMAT number properties note: However, there are no multiples of 7 among the known values.


GMAT statistics note: Watch divisibility, factors, remainders, parity, and prime structure. Missing fourth value must be a multiple of 7.


GMAT statistics note: The only multiple of 7 between 3 and 13 is 7 itself, so this is the missing integer and the mean can now be calculated. The four known integers sum to 19.


GMAT statistics note: Since the unknown integer must be positive, the sum of the entire set must be greater than 19. The next multiples of 13 are 13(2) = 26 and 13(3) = 39, which would imply that the missing integer is 26 – 19 = 7 or 39 – 19 = 20, respectively.


GMAT statistics note: (For greater multiples of 13, the missing integer would have to be even greater than 20.) However, the missing integer must be between 3 and 13, so 20 is too great. The missing integer must be 7, and the mean can be calculated.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If , which of the following is true of g?


GMAT number properties note: If x and y are positive integers, what is the remainder when xyis divided by 10? For all positive integers n, the sequence An is defined by the following relationship.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. What is the sum of all the terms in the sequence from A1 through A10, inclusive?


GMAT number properties note: If x and y are positive integers and n = 5x + 7y + 3, what is the units digit of n? Y = 2x − 16(1) y is divisible by 4.(2).


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. At a certain store, all notebooks have the same price and all pencils have the same price.


GMAT sequences and patterns note: The price of four notebooks and three pencils is more than $12 and less than $13. The price of two notebooks and five pencils is more than $8 and less than $9.


GMAT sequences and patterns note: If a notebook costs x more than a pencil, which of the following could be the value of x? A certain sequence is defined by the following rule:.


GMAT sequences and patterns note: Sn = k(Sn − 1), where k is a constant. If S1 = 2 and S13 = 72, what is the value of S7?


GMAT sequences and patterns note: N is divisible by 3.(1) n is odd.(2).


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. In a certain sequence, each term a er the first is twice the previous term.


GMAT sequences and patterns note: If the first term of the sequence is 3, what is the sum of the 14th, 15th, and 16th terms in the sequence? For n > 0, the sequence tn is defined as tn = tn − 1 + n.


GMAT sequences and patterns note: N + 1 is divisible by 3.(1) n − 1 is divisible by 4.(2) 30. To complete a 120-mile trip, a train first travels at a constant rate of x miles per hour (mph) for 80 miles.


GMAT sequences and patterns note: Then it travels the remaining 40 miles at a constant rate of y mph. If the train had instead completed the entire trip at a constant rate of z mph, the trip would have taken 1 hour longer.


GMAT sequences and patterns note: Which of the following is the value of z in terms of x and y?


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Therefore, f(x) equals the average of the integers from 1 to x, which is , multiplied by the number of integers from 1 to x, which is x.


GMAT number properties note: Therefore, f(x) = The question asks for the sum of the values of , and so on up to . Instead of summing these values by hand, look for a pattern, since the problem only asks for a range and not a specific answer.


GMAT number properties note: One possibility is to use algebra. Can be simplified as follows.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Using this formula, the first few terms of the sum are , and so on, until the tenth term, which is .


GMAT sequences and patterns note: All of these terms are greater than and no greater than 1. If all of the terms were 0.5, then the sum would be 10(0.5) = 5.


GMAT sequences and patterns note: However, if all of the terms were 1, then the sum would be 10. Therefore, the sum must be between 5 and 10.


GMAT sequences and patterns note: This problem can also be solved without using a formula for f(x). Manually calculate the first few values of f(x), and look for a pattern.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. The values appear to be decreasing and approaching .


GMAT sequences and patterns note: If each term in the sum is greater than but no greater than 1, their sum will be greater than 5 but no greater than 10. The question asks for the remainder when xy is divided by 10.


GMAT sequences and patterns note: The remainder of any number when divided by 10 is equal to the units digit of that number, so the question is really asking for the units digit of xy. But 26 has a units digit of 6, and remember that 6any positive integer has a units digit of 6 (the pattern is a single-term repeat).


GMAT sequences and patterns note: Thus, 26 raised to ANY positive integer power will also have a units digit of 6 and therefore a remainder of 6 when divided by 10. Given that yx = 1, there are a few possible scenarios.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. X y yx = 1 0 anything nonzero (anything nonzero)0 = 1 ✓ anything 1 1anything = 1 ✓ even −1 −1even = 1 ✓ However, the question stem indicates that x and y are POSITIVE integers, so eliminate the first and third scenarios.


GMAT number properties note: The remaining scenario indicates that y = 1 and x = any positive integer. Without more information about x, you cannot determine the remainder when xy is divided by 10.


GMAT number properties note: Set up a table to list the first few terms of the sequence and also the cumulative sum.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. N Cumulative Sumn Cumulative Sum As you build the table, compare the input column values (n) with the output column values (cumulative sum), looking for a pattern.


GMAT sequences and patterns note: The denominator of the cumulative sum is n! And the numerator is one less than n!.


GMAT sequences and patterns note: Sum of terms through Substitute to find the sum through A10.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. The units digit of n is determined solely by the units digit of the expressions 5x and 7y + 3 because when two numbers are added together, the units digit of the sum is determined solely by the units digits of the added numbers.


GMAT sequences and patterns note: Since x is a positive integer, and 5any positive integer always has a units digit of 5, 5x always ends in a 5. However, the units digit of 7y + 3 is not certain, as the units digit pattern for the powers of 7 is a four-term repeat: [7, 9, The question can be rephrased as, “What is the units digit of 7y + 3?” Note:.


GMAT sequences and patterns note: Determining y would be one way of answering the question above, but don’t rephrase to, “What is y?” Because the units digits of the powers of 7 have a repeating pattern, you might get a single answer for the units digit of 7y + 3 despite having multiple values for y.


GMAT sequences and patterns note: This statement indicates neither the value of y nor the units digit of 7y + 3, as y depends on the value of x, which could be any positive integer. By contrast, if x = 10, then y = 4, and 7y + 3 has a units digit of 3.


GMAT sequences and patterns note: The statement does indicate that y is an even number, so the exponent y + 3 must be odd. As a result, there are only two possible values for the units digit of the d.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Y y + 3 Units Digit of 7y + 3y y + 3 Units Digit of 7y + 3 Ultimately, this means that n has a units digit of 5 + 3 = 8.


GMAT sequences and patterns note: Instead, start by translating the information in the problem into math. The first piece of information in the problem translates to the following inequality:.


GMAT sequences and patterns note: The second piece of information translates to this inequality.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. The question asks for the difference between the cost of a notebook and the cost of a pencil.


GMAT statistics note: In other words, what is the value of n – p? One option is to use the previous two inequalities to create a third inequality, which includes the combo n – p.


GMAT statistics note: One way to accomplish this is to multiply the second inequality by –1, taking care to flip the signs. Then, reorder the result so that the inequality signs point in the expected direction:.


GMAT statistics note: Now, sum the two inequalities. You can add inequalities as long as the inequality signs are facing the same direction:.


GMAT statistics note: This simplifies to 1.5 < n – p < 2.5. It is also possible to “solve” the two inequalities for n and for p, separately, then to use the resulting ranges to find the possible differences between n and p.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. To form each new term of the sequence, multiply the previous term by k.


GMAT sequences and patterns note: If S1 = 2, then S2 = 2k, and S3 = 2k2, and Sn = 2kn − 1. Since the problem indicates that S13 = 72, set up an equation to solve for k.


GMAT sequences and patterns note: Plug the value of k into that expression to calculate S7. Remember the units digit pattern for 9x, where x is an integer.


GMAT sequences and patterns note: The units digit of 9x is 9 if x is odd, but the units digit is 1 if x is even: a repeating pattern of [9, 1]. Now, consider the sums of the powers of 9 up to 9n.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. 1 1 + 9 = 10 0 2 1 + 9 + 81 = 91 1 3 [1 + 9 + 81] + 729 = [units digit of 1] + units digit of 9 0 4 Units digit of 0 + units digit of 1 1 Odd Units digit: pairs of (1 + 9) 0 Even Units digit: pairs of (1 + 9) + 1 1 The alternating 1’s and 9’s in the units digits pair to a sum of 10, or a units digit of 0.


GMAT sequences and patterns note: Thus, the units digit of the sum displays another two- term repeating pattern. The units digit of y is 0 if n is odd, but 1 if n is even.


GMAT sequences and patterns note: The remainder when y is divided by 5 depends only on the units digit and will be either 0 or 1 as well. The rephrased question is, “Is n odd or even?” (1) INSUFFICIENT:.


GMAT sequences and patterns note: If n is a multiple of 3, it may be either odd or even. If n is odd, the units digit of y is 0, and the remainder is 0 when y is divided by 5.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. The sequence is 3, 6, 12, 24, 48, and so on.


GMAT sequences and patterns note: You could write out the first 16 terms and add the 14th, 15th, and 16th together, but such an approach would be prone to error and time-consuming. Write the sequence in terms of the powers of 2: 3, 6, 12, 24, 48, and so on 3(20), 3(21), 3(22), 3(23), 3(24), and so on So Sn, the nth term of the sequence, equals 3(2n − 1).


GMAT sequences and patterns note: Thus, the sum of the 14th, 15th, and 16th terms equals 3(213) + 3(214) + All of the terms share the common factors 3 and 213, so factor those terms out.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. Sequence problems are o en best approached by charting out the first several terms of the given sequence.


GMAT sequences and patterns note: In this case, keep track of n, tn, and whether tn is even or odd. 0 3 Odd 1 3 + 1 = 4 Even 2 4 + 2 = 6 Even 3 6 + 3 = 9 Odd 4 9 + 4 = 13 Odd 5 13 + 5 = 18 Even 6 18 + 6 = 24 Even 7 24 + 7 = 31 Odd 8 31 + 8 = 39 Odd Notice that beginning with n = 1, a four-term repeating cycle of [even, even, odd, odd] emerges for tn.


GMAT sequences and patterns note: Thus, a statement will be sufficient only.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. If it indicates how n relates to a multiple of 4 (i.e., n = a multiple of 4 ± known constant).


GMAT algebra note: This statement does not indicate how n relates to a multiple of 4. If n + 1 is a multiple of 3, then n + 1 could be 3, 6, 9, 12, 15, etc.


GMAT algebra note: This means that n could be 2, 5, 8, 11, 14, etc. From the chart, if n = 2 or n = 5, then tn is even.


GMAT algebra note: However, if n = 8 or n = 11, then tn is odd. This statement indicates exactly how n relates to a multiple of 4.


GMAT algebra note: If n − 1 is a multiple of 4, then n − 1 could be 4, 8, 12, 16, 20, etc., and n could be 5, 9, 13, 17, 21, etc. From the chart (and the continuation of the four-term pattern), tn must be even.


GMAT algebra note: If you choose to use Smart Numbers, select your numbers cautiously to simplify the arithmetic. Choose x = y = 40, so that the train completes the entire trip at 40 mph, taking = 3 hours.


GMAT algebra note: If the train had instead traveled at z mph, it would have taken one more hour, or 4 hours in total.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. The correct answer will yield a value of z = 30 mph when x = 40 and y = 40 are plugged in.


GMAT word problem note: Alternatively, solve algebraically by creating an RTD chart. Rate Time Distance x mph hours 80 miles.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Rate Time Distance y mph hours 40 miles (Original speed) hours 120 miles z mph hours 120 miles Use the last row of the chart to create an equation, which can then be solved for z.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. If y ≠ x, then is the equivalent of which of the following?


GMAT sequences and patterns note: In the sequence gn defined for all positive integer values of n, g1 = g2 = 1 and, for n ≥ 3, gn = gn − 1 + 2n − 3. If the function equals the sum of the terms g1, g2...


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If 3k + 3k = (39)39 − 3k, then what is the value of k?


GMAT number properties note: If x and y are positive integers, what is the value of xy?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If k is an integer and is divisible by 6k, what is the maximum possible value of k?


GMAT number properties note: In a certain sequence, the term Sn is given by the formula Sn = (n + 1)! Which of the following is equivalent to the difference between S100 and S99?


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Simplify the right side of the equation.


GMAT algebra note: This still needs to be manipulated in order to match the le side of the equation, since the bases need to match their exponents. The number 33 is okay, but 28 is not.


GMAT algebra note: Try turning it into a base of 4 instead. That’s it: x = 3 and y = 4, since the problem states that y > x.


GMAT algebra note: Note that the expression can be rewritten as follows by pairing up consecutive terms in the sequence. In each of these pairs, the 94 term cancels out.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Therefore, the value of K is as follows.


GMAT algebra note: First, rephrase the question stem by subtracting xy from both sides: “Is xy < 0?” The question is whether xy is negative, or “Do x and y have opposite signs?” Be careful! Dividing by variables is the mistake:.


GMAT algebra note: Incorrect as a result of the mistake in the previous step.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Not only does this rephrase make the statements moot (2 is definitely not less than 1, no matter what the statements say), but it also ignores some special cases.


GMAT fractions, decimals, and percents note: If xy = 0, then dividing by xy yields an undefined value. If xy < 0, you should have flipped the sign of the inequality.


GMAT fractions, decimals, and percents note: Instead, use the correct rephrasing: “Do x and y have opposite signs?” (1) INSUFFICIENT:. If , then x2 and y must have opposite signs.


GMAT fractions, decimals, and percents note: Since x2 must be positive, y must be negative. However, x could be either positive or negative.


GMAT fractions, decimals, and percents note: If x3y3 < (xy)2, then you can divide both sides by (xy)2 (since that quantity is positive). The simplified inequality is xy < 1, which is not sufficient to answer the question.


GMAT fractions, decimals, and percents note: Each statement indicates that xy could be either positive or negative. The statements are equally insufficient and neither provides any additional information to the other.


GMAT fractions, decimals, and percents note: First, distribute the numerator. Group the numerator terms with x and −y in mind.


GMAT strategy note: Alternatively, you could choose smart numbers. Plug the selected values into the choices.


GMAT strategy note: The choice that equals 7 is the correct answer. Begin by listing some values of gn to get a sense for how gn progresses.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Now, look for a pattern in the sums defined by Each value is double the previous value:.


GMAT algebra note: You have already discovered that g3 = 2, so you can select g3 as the answer. The common term in this problem is the recurring base of 3.


GMAT algebra note: Group like terms (i.e., all the terms with k on the le side, all the other.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Powers of 3 on the right side), then simplify each power of 3 using exponent rules.


GMAT number properties note: Note the constraints that x and y are positive integers. If y = 5 and x = 6, then this statement is true: 6!


GMAT number properties note: Are these the only possible positive integer values for x and y? The constant 6 is equal to 3!, so there’s one more: when y = 1 and x = 3, the statement is also true: 3!


GMAT number properties note: You can multiply both sides by y to eliminate that variable entirely. In other words, this statement doesn’t indicate anything about y.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The quadratic produces two solutions, but only x = 6 is valid, since x must be a positive integer.


GMAT number properties note: This is sufficient to calculate a value for xy. The question asks for the greatest value of k such that is divisible by 6k.


GMAT number properties note: Since 6 = 3 × 2, the greatest value of k will equal the number of 3 × 2 pairs among the prime factors of.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. To count the number of times 3 appears as a factor of , rewrite the expression, pulling out any factor(s) of 3 from each term.


GMAT number properties note: There are six factors of 3 in To count the number of times 2 appears as a factor of , rewrite the expression, pulling out any factor(s) of 2 from each term. = (33)(25)(31)(2 × 15)(29)(22 × 7)(27)(2 × 13)(25)(23 ×3)(23) There are twelve factors of 2 in Since there are twelve 2’s but only six 3’s, there are only six 3 × 2 pairs among the prime factors of .


GMAT number properties note: In general, focus on the greatest prime in the divisor (in this case, the six maximum possible factors of 3 in 6k), as it will be the limiting factor.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. This is one of the incorrect answers.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If x and y are positive integers, what is the value of x2 = 2xy − y2(1) 2xy = 8(2) 42.


GMAT number properties note: What is the remainder when (47)(49) is divided by 8? If a = 4x2 + 4xy and b = 4y2 + 4xy, which of the following is equivalent to x + y?


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If x is the square of an integer, is y the square of an integer?


GMAT number properties note: Xy is the square of an integer.(1) is the square of an integer.(2) 47. If x and y are positive integers such that x > y and , which of the following is equivalent to 2b?


GMAT quant note: If , what is the value of xy?


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Factor this “square of a difference” special product and simplify.


GMAT quant note: Since x = y, the value of is 1. Test different values for x and y.


GMAT quant note: X (positive) y (positive) xy = 4 2 2 2 × 2 = 4 ✓ 1.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. X (positive) y (positive) xy = 4 1 4 1 × 4 = 4 ✓ Alternatively, manipulate the equation xy = 4 to get on one side.


GMAT algebra note: The expression does not equal a constant, so could take on many different values. One way to solve would be to multiply (47)(49), then either divide the result by 8 or repeatedly subtract known multiples of 8 from the result until you are le with a remainder less than 8.


GMAT algebra note: An alternative is to rewrite the given product as an equivalent yet easier- to-manipulate product. Note that 47 and 49 are equidistant from 48, a multiple of 8.


GMAT algebra note: Write each of the original factors as terms in the form (a + b) or (a − b).


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. (47)(49) = (48 + 1)(48 − 1) This form is the difference of two squares special product, (a + b)(a − b) = a2 − b2.


GMAT quant note: (48 + 1)(48 − 1) = (482 − 12) Forty-eight is a multiple of 8, and therefore so is 482. Thus, (482 − 12) is 1 less than a multiple of 8.


GMAT quant note: All such numbers (e.g., 7, 15, 23, 31) have a remainder of 7 when divided by 8. Add a and b to get a + b = 4x2 + 8xy + 4y2 = 4(x2 + 2xy + y2).


GMAT quant note: The right side is the square of a sum, so factor and solve. Note that you could safely take the square root of both sides because any square is non-negative.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Alternatively, choose smart numbers.


GMAT algebra note: For values to plug into the choices, first compute a and b. The one that equals 5 is the correct answer.


GMAT algebra note: Manipulate the question expression, noting the special products in the exponents.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. The rephrased question is, “What is the value of ab?” Knowing the values of a and b individually would be sufficient, of course, but the individual values are not required as long as you can determine ab.


GMAT quant note: It is impossible to manipulate a + b = 7 to get ab, nor can you solve for a and b individually. This statement answers the rephrased question directly.


GMAT quant note: Notice the special product of the form (a + b)(a − b) = a2 − b2 under the square root symbol. Jot down the information provided in the question stem.


GMAT quant note: If x is the square of an integer, then x = i2, where i is an unknown integer.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. (1) INSUFFICIENT: x is a perfect square, and when multiplied by the unknown value y, it turns into a (possibly different) perfect square.


GMAT quant note: This situation can occur if y is also a perfect square. Since y is the square of an integer, the answer is Yes.


GMAT quant note: If you only consider integer values for y, it seems as though y must be a square. For instance, if x = 22, y cannot equal 2, 3, 5, 6, or any non-square, since 2(22), 3(22), 5(22), etc., are not squares.


GMAT quant note: However, the question does not restrict y to integer values. To find a non- integer value of y that fits the known information, start by choosing square values for both x and xy.


GMAT quant note: In this case, y = , which is not a perfect square. Therefore, the answer is No.


GMAT quant note: (2) SUFFICIENT: is the square of an integer, so = j2, for some unknown integer j. From the question stem, x = i2, for some unknown integer i.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Therefore, y must be the square of an integer.


GMAT algebra note: The answer is definitely Yes, and this statement is sufficient. To solve this problem, isolate b in the equation.


GMAT algebra note: Most of the choices are not fractions, so try to cancel the denominator. Since x − y is a well-disguised difference of two squares, factor the numerator and denominator.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Each of the expressions given is equal to 16, so set them equal to each other.


GMAT algebra note: Note that you have the square of a sum and the difference of two squares special products. Put them both in distributed form, then simplify.


GMAT algebra note: Since 2y(x + y) = 0, it must be true that either 2y or (x + y) is equal to 0. However, (x + y) cannot equal 0, since the problem indicates that (x + y)2 = 16.


GMAT algebra note: So it must be that 2y = 0, and therefore y = 0. Plug in 0 for y in x2 – y2 = 16 to get x2 = 16.


GMAT algebra note: Use this to solve for y in the second equation.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Do x = ±4 and y = 0 also work in the first equation?


GMAT sequences and patterns note: There are several ways to approach this problem without extensive calculation. The units digit of 5,9952 must equal the units digit of 52, which is 5.


GMAT sequences and patterns note: Similarly, the units digit of 4,7962 must equal the units digit of 62, which is 6. Subtracting a number with a units digit of 6 from a number with a units digit of 5 results in a units digit of 9.


GMAT sequences and patterns note: Estimation It is also possible to estimate the answer: 5,995 is approximately equal to 6,000, and 4,796 is approximately equal to 5,000.