Competition problem — Number theory

Library / /337 of 520

Number theory Difficulty 6.3 National olympiad Prove it

Theorem 2 Let $a, b$ be two given integers, $a \neq 0$ ; and let $d$ be a given integer. Then, there certainly exists a unique pair of integers $q_{1}$ and $r_{1}$ , satisfying
$b=q_{1} a+r_{1}, \quad d \leqslant r_{1}<|a|+d .$

Moreover, $a \mid b$ if and only if $a \mid r_{1}$ .

Solution

None

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