Prove that if $\\gcd(a,b)=1$ then $a\mathbb{N}\cap b\mathbb{N}=(ab)\mathbb{N}.$

Question

Prove that if $\\gcd(a,b)=1$ then $a\mathbb{N}\cap b\mathbb{N}=(ab)\mathbb{N}.$

We have $1=\gcd(a,b)\implies au+bv=1$ for some $u,v\in \mathbb{Z}.$ Is it the correct way to prove the above result?

I am stuck here.

Answer 1

Can you show that $(ab)\mathbb N$ is a subset of both $a\mathbb N$ and $b\mathbb N?$ So $(ab)\mathbb N\subseteq a\mathbb N\cap b\mathbb N?$

Then use $au+bv=1$ to show that if $k\in a\mathbb N\cap b\mathbb N$ then $k\in ab\mathbb N.$