2007 AMC 10A Problem 17

Question

Suppose that

m

and

n

are positive integers such that

75m = n^3.

What is the minimum possible value of

m + n?

15

30

50

60

5700

In a perfect cube, every prime’s exponent is a multiple of

3

Since

75 = 3 \cdot 5^2,

find the smallest

m

making all exponents multiples of

3

Since

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

every prime factor must occur a multiple of three times.

The smallest such

m

is

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.

Problem 17 in other years