Track / Stage 5 / 343 of 400 #943 of 1964

1. 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.

I solved it I didn't Skip

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 and solution reproduced as published; topic, difficulty and ordering added by this site.