1977 AMC 12 第 14 题

Question

有多少个整数对

(m,n)

满足方程

m+n=mn

？

How many pairs

(m,n)

of integers satisfy the equation

m+n=mn?

1

2

3

4

多于

4

个

more than

4

把所有项移到等式一边，再加上

1

Move all terms to one side and add

1

将方程化为两个整数的乘积等于

1

Factor the equation into a product of two integers equal to

1

移项并配成乘积，得到

\begin{aligned} mn-m-n&=0,\\ (m-1)(n-1)&=1 \end{aligned}\text{。}

1

的整数因数对为

(1,1)

和

(-1,-1)

，分别得到

(m,n)=(2,2)

和

(0,0)

。因此共有两个有序整数对。

因此，正确答案是 B。

Rearranging and completing the product gives

\begin{aligned} mn-m-n&=0,\\ (m-1)(n-1)&=1. \end{aligned}

The integer factor pairs of

1

are

(1,1)

and

(-1,-1),

producing

(m,n)=(2,2)

and

(0,0).

Thus there are two ordered pairs.

Therefore, the correct answer is B.

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