Divides at least one of them?

Number Theory Level 2

True or False?

$\quad$ Let $a$ , $b$ and $c$ be three positive integers. Then,
$\quad$ $a|bc$ implies $a|b$ or $a|c$ .

Notation: Here $a|b$ means, $a$ divides $b$ , that is, there exists another integer $c$ such that $b=ac$ .

True False

1 solution

Chew-Seong Cheong
Nov 29, 2016

Let $a=6$ , $b=4$ and $c=9$ . We note that $6 \nmid 4 \implies a \nmid b$ and $6 \nmid 9 \implies a \nmid c$ but $6 | 36 \implies a | bc$ .