AMC10 2012 B
AMC10 2012 B · Q10
AMC10 2012 B · Q10. It mainly tests Basic counting (rules of product/sum), Divisibility & factors.
How many ordered pairs of positive integers $(M, N)$ satisfy the equation $\frac{M}{6} = \frac{6}{N}$?
有几个正整数有序对 $(M, N)$ 满足方程 $\frac{M}{6} = \frac{6}{N}$？
(A)
6
6
(B)
7
7
(C)
8
8
(D)
9
9
(E)
10
10
Answer
Correct choice: (D)
正确答案：（D）
Solution
Multiplying the given equation by $6N$ gives $MN = 36$. The\ndivisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Each of these divisors can be\npaired with a divisor to make a product of 36. Hence there are 9 ordered pairs\n$(M, N)$.
将给定方程两边乘以 $6N$ 得 $MN = 36$。36 的因数是 1、2、3、4、6、9、12、18 和 36。这些因数中的每一个都可以与另一个因数配对使积为 36。因此有 9 个有序对 $(M, N)$。
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