1977 AMC 12 Problem 14

Attempt Problem 14 of the 1977 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1977 AMC 12 solutions, or check the answer key.

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14.

How many pairs

(m,n)

of integers satisfy the equation

m+n=mn?

1

2

3

4

more than

4

Move all terms to one side and add

1

Factor the equation into a product of two integers equal to

1

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.

Problem 14 in other years