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"value": "Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?\nPlease reason step by step, and put your final answer wi... |
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"value": "Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?\nPlease reason step by step, and put your final answer wi... |
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"value": "Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?\nPlease reason step by step, and put your final answer wi... |
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"from": "user",
"value": "Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?\nPlease reason step by step, and put your final answer wi... |
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"value": "Find the remainder when $9 \\times 99 \\times 999 \\times \\cdots \\times \\underbrace{99\\cdots9}_{\\text{999 9's}}$ is divided by $1000$ .\nPlease reason step by step, and put your final answer within \\boxed{}."
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"value": "<think>\nOkay, so... |
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"from": "user",
"value": "Find the remainder when $9 \\times 99 \\times 999 \\times \\cdots \\times \\underbrace{99\\cdots9}_{\\text{999 9's}}$ is divided by $1000$ .\nPlease reason step by step, and put your final answer within \\boxed{}."
},
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"from": "assistant",
"value": "<think>\nOkay, le... |
[
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"from": "user",
"value": "Find the remainder when $9 \\times 99 \\times 999 \\times \\cdots \\times \\underbrace{99\\cdots9}_{\\text{999 9's}}$ is divided by $1000$ .\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "assistant",
"value": "<think>\nOkay, le... |
[
{
"from": "user",
"value": "Find the remainder when $9 \\times 99 \\times 999 \\times \\cdots \\times \\underbrace{99\\cdots9}_{\\text{999 9's}}$ is divided by $1000$ .\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "assistant",
"value": "<think>\nOkay, so... |
[
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"from": "user",
"value": "One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$ . Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel... |
[
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"from": "user",
"value": "One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$ . Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel... |
[
{
"from": "user",
"value": "One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$ . Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel... |
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