2007 AMC 10A Problem 17 — Solution | LIVE by Po-Shen Loh

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2007 AMC 10A Problem 17

Below is the professionally curated solution for Problem 17 of the 2007 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2007 AMC 10A solutions, or check the answer key.

All of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:perfect power prime factorization

Difficulty rating: 1480

17.

Suppose that $m$ and $n$ are positive integers such that $75m = n^3.$ What is the minimum possible value of $m + n?$

$15$

$30$

$50$

$60$

$5700$

Solution:

Since $n^3 = 75m = 3 \cdot 5^2 \cdot m,$ every prime factor must occur a multiple of three times.

The smallest such $m$ 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.

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