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1977 AHSME Problems/Problem 14

Revision as of 13:19, 21 November 2016 by E power pi times i (talk | contribs) (Created page with "== Problem 14 == How many pairs <math>(m,n)</math> of integers satisfy the equation <math>m+n=mn</math>? <math>\textbf{(A) }1\qquad \textbf{(B) }2\qquad \textbf{(C) }3\qquad...")
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Problem 14

How many pairs $(m,n)$ of integers satisfy the equation $m+n=mn$?

$\textbf{(A) }1\qquad \textbf{(B) }2\qquad \textbf{(C) }3\qquad \textbf{(D) }4\qquad \textbf{(E) }\text{more than }4$

Solution

Solution by e_power_pi_times_i

If $m+n=mn$, $mn-m-n = (m-1)(n-1)-1 = 0$. Then $(m-1)(n-1) = 1$, and $(m,n) = (2,2) or (0,0)$. The answer is $\boxed{\textbf{(B) }2}$.

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