AMC10 2007 A

AMC10 2007 A · Q17

AMC10 2007 A · Q17. It mainly tests Primes & prime factorization.

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

假设 $m$ 和 $n$ 是正整数使得 $75m = n^3$。$m + n$ 的最小可能值是多少？

(A) 15 15

(B) 30 30

(D) 60 60

(E) 5700 5700

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): An integer is a cube if and only if, in the prime factorization of the number, each prime factor occurs a multiple of three times. Because $n^3 = 75m = 3 \cdot 5^2 \cdot m$, the minimum value for $m$ is $3^2 \cdot 5 = 45$. In that case $n = 15$, and $m + n = 60$.

答案（D）：一个整数是完全立方数，当且仅当在它的质因数分解中，每个质因数出现的次数都是 3 的倍数。因为 $n^3 = 75m = 3 \cdot 5^2 \cdot m$，所以 $m$ 的最小值是 $3^2 \cdot 5 = 45$。此时 $n = 15$，并且 $m + n = 60$。

Topics

NT.002 Primes & prime factorization