Competition problem — Number theory

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Number theory Difficulty 4.9 AIME Prove it Estonia

A positive integer $n$ is interesting, if for some positive integer $m$ and positive integers $a$ , $b$ that are smaller than $m$ , $\frac{m^2}{ab} = n$ . For example, $10$ is interesting because $\frac{20^2}{4 \cdot 10} = 10$ . Find the smallest interesting integer.

Solution

For $n = 2$ we can take $m = 12$ , $a = 8$ and $b = 9$ , because $\frac{12^2}{8 \cdot 9} = \frac{144}{72} = 2$ .

On the other hand, $1$ is not interesting, because if $\frac{m^2}{ab} = 1$ , or $m^2 = ab$ , then $a$ and $b$ cannot both be less than $m$ at the same time.

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