Library / Estonia — Open Contests /13 of 63

Estonia — Open Contests

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 $15$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

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