Track / Stage 4 / 320 of 340 #1060 of 2444

Problem 1060

AMC 12 late, AIME early

Number theory Difficulty 5.0 Prove it Ukrainian National Mathematical Olympiad, 4th Round · Ukraine

Let $a, b, c, d$ be distinct natural numbers such that $ab + cd$ is divisible by $ac + bd$ . Prove that $ac + bd$ is a composite number.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

To the contrary, assume that $ac + bd$ is prime. Then $ac + bd \mid ab + cd \Rightarrow ac + bd \mid ac + bd + ab + cd = (a + d)(b + c)$ , and so $ac + bd \mid a + d$ or $ac + bd \mid b + c$ . But this is impossible because for distinct $a, b, c, d$ we have that $ac + bd > a + d$ and $ac + bd > b + c$ . This contradiction completes the proof.

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