Competition problem — Number theory

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Number theory Difficulty 4.9 AIME Prove it Estonia

Juku conjectured the following in his mathematics circle: whenever the product of two coprime integers $x$ and $y$ is divisible by the product of some two coprime integers $a$ and $b$ , at least one of $x$ and $y$ is divisible by $a$ or $b$ . Does his proposition hold?

Solution

Let $x = 20$ , $y = 21$ , $a = 14$ , $b = 15$ . Then $x$ and $y$ are coprime, as they are consecutive, similarly $a$ and $b$ are coprime. The product $xy = 420$ is divisible by $ab = 210$ but neither of $20$ and $21$ is divisible by $14$ or $15$ .

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