2002 AMC 12A Problems/Problem 6

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Problem

For how many positive integers $m$ does there exist at least one positive integer n such that $m \cdot n \le m + n$?

$\mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 12\qquad \mathrm{(E) \ }$ infinitely many


Solution

$m \cdot 1 \le m + 1$ for $n=1$

$m \le m + 1$

$\Rightarrow \mathrm{(E) \ }$ infinitely many