# Math Help - Divisibility proof

1. ## Divisibility proof

gcd(a,b) = 1 and a|m and b|m. How to prove ab|m without using unique-prime-factorization theorem?

2. Hello,
Originally Posted by rf0
gcd(a,b) = 1 and a|m and b|m. How to prove ab|m without using unique-prime-factorization theorem?
$a \mid m \implies m=am' ~,~ \text{where } m' \in \mathbb{Z}$

$b \mid m \implies m=bm'' ~,~ \text{where } m'' \in \mathbb{Z}$

Therefore $am'=bm''$ and hence a divides $bm''$

By Gauss theorem, if $\text{gcd}(a,b)=1$ and $a \mid bc$, then $a \mid c$
So here, we can conclude that $a \mid m''$.
So we can write $m''=am''' ~,~ \text{where } m''' \in \mathbb{Z}$

So $m=bm''=b(am''')=(ab)m'''$

Therefore $ab \mid m \quad \quad \square$

3. Thanks a bunch, I understand it now