Question

有多少个正整数有序对 $(M,N)$ 满足？ $\frac{M}{6}=\frac{6}{N}?$

How many ordered pairs of positive integers $(M,N)$ satisfy the equation $\frac{M}{6}=\frac{6}{N}?$

$6$

$7$

$8$

$9$

$10$

交叉相乘得 $MN = 36$ 。因此可让 $M$ 取 $36$ 的任一正因数，再唯一确定 $N$ 。

由于 $36=2^2\cdot 3^2$ ，共有 $(2+1)(2+1)=9$ 个可能的 $M$ ，每个也对应唯一的 $N$ 。

所以正确答案是 D。

By cross multiplying, we can see that $MN = 36.$ Thus, we can make $M$ any factor of $36$ and then determine $N$ from it.

Since $36=2^2\cdot 3^2,$ we have $(2+1)(2+1)=9$ possible choices for $M,$ each of which also determine a unique $N.$

Thus, the correct answer is D .