2007 iTest Problems/Problem 24
Problem
Let be the smallest positive integer such that is a perfect square and is a perfect cube. Find the remainder when is divided by .
Solution
The prime factorization of is , and the prime factorization of is . Since is a perfect square and is a perfect cube, the exponents in the prime factorization of are even, and the exponents in the prime factorization of are a multiple of three.
To make the exponents of even, the exponent of in is at least , but since there are no powers of in , the exponent of in is at least . Similarly, in , the exponent of is at least , the exponent of is at least , and the exponent of is at least .
Thus, , the minimum positive integer that satisfies the criteria, equals . Using modular arithmetic to find the remainder, The remainder when is divided by is .
See Also
2007 iTest (Problems, Answer Key) | ||
Preceded by: Problem 23 |
Followed by: Problem 25 | |
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