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Harvard-MIT Mathematics Tournament

Algebra Difficulty 4.9 AIME Prove it United States

Problem:
A broken calculator has the $+$ and $\times$ keys switched. For how many ordered pairs $(a, b)$ of integers will it correctly calculate $a+b$ using the labelled $+$ key?

Solution

Solution:
We need $a + b = a b$ , or $a = \frac{b}{b-1} = 1 - \frac{1}{b-1}$ , so $1/(b-1)$ is an integer. Thus $b$ must be $0$ or $2$ , and $a$ is $0$ or $2$ , respectively. So there are $2$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.