2010 AMC 12B Problems/Problem 9
Problem
Let be the smallest positive integer such that is divisible by , is a perfect cube, and is a perfect square. What is the number of digits of ?
Solution
We know that and . Cubing and squaring the equalities respectively gives . Let . Now we know must be a perfect -th power because , which means that must be a perfect -th power. The smallest number whose sixth power is a multiple of is , because the only prime factors of are and , and . Therefore our is equal to number , with digits .
Solution 2 (Chinese Remainder Theorem)
Let for some integer , then we know that
This means we can write and thus we have
, therefore we know
Thus, our answer is .
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See also
2010 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
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All AMC 12 Problems and Solutions |
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