Question

设 $m$ 和 $n$ 是正整数，且 $75m = n^3$ 。 $m + n$ 的最小可能值是多少？

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

$15$

$30$

$50$

$60$

$5700$

因为 $n^3 = 75m = 3 \cdot 5^2 \cdot m$ ，每个质因数的指数都必须是三的倍数。

最小这样的 $m$ 是 $3^2 \cdot 5 = 45$ ，此时 $n^3 = 3^3 \cdot 5^3$ ，所以 $n = 15$ 。

因此 $m + n = 45 + 15 = 60$ 。

所以正确答案是 D。

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.