Track / Stage 5 / 394 of 400 #994 of 1964

Problem 994

AIME late

Number theory Difficulty 5.8 Prove it

Let $a, b$ be integers, $a \geqslant 1, b=q a+r, 0 \leqslant r<a$ . Prove:
$q=[b / a], \quad r=a\{b / a\} .$

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

1. $b / a=[b / a]+\{b / a\}, b=[b / a] a+\{b / a\} a$ , This gives a new proof of the division algorithm (Theorem 1, §3).

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.