Question

三个正整数都大于 $1$ ，乘积为 $27000$ ，且两两互质。它们的和是多少？

Three positive integers are each greater than $1,$ have a product of $27000 ,$ and are pairwise relatively prime. What is their sum?

$100$

$137$

$156$

$160$

$165$

分解后可知质因数只有 $2$ 、 $3$ 、 $5$ 。因为三个数两两互质，每个素数幂只能完整属于其中一个数。

又因为三个数都大于 $1$ ，所以每个数都必须含有其中一种质因数。

因为所以三个数为 $2^3,3^3,5^3$ ，和为 $160$ 。 $27000=2^3\cdot 3^3\cdot 5^3,$

所以正确答案是 D。

Only one of them is a multiple of $2,$ only one of them is a multiple of $3,$ and only one of them is a multiple of $5.$

Since each of the positive integers is greater than $1,$ each of them must be a multiple of one of the given primes.

Therefore, since $27000=2^3\cdot 3^3\cdot 5^3,$ the numbers must be $2^3,3^3,5^3,$ making their sum $160.$

Thus, the correct answer is D .