2007 AMC 10A Problem 17

Problem 17 of 25IntermediateNumber Theory

Suppose that

m

and

n

are positive integers such that

75m = n^3.

What is the minimum possible value of

m + n?

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Solution

Since

n^3 = 75m = 3 \cdot 5^2 \cdot m,

every prime factor must occur a multiple of three times. The smallest such

m

3^2 \cdot 5 = 45,

giving

n^3 = 3^3 \cdot 5^3

and

n = 15.

Then

m + n = 45 + 15 = 60.

Thus, the correct answer is D.

Concepts: perfect power · prime factorization

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