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

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2007 AMC 10A Problem 17

Attempt Problem 17 of the 2007 AMC 10A below, then check your answer against the professionally curated solution 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 problems are used with official legal permission of the Mathematical Association of America (MAA).

17.

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

$15$

$30$

$50$

$60$

$5700$

Answer: D

Concepts:perfect power prime factorization

Difficulty rating: 1480

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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