Prove $\,\gcd(a,b)\mid {\rm lcm}(a,b)\ $ I have a hard time with this question, can someone explain?
&quot;Prove that for any number $a$ and $b$, $(a,b)$ goes into $\text{lcm}(a;b)$&quot;
Hint: by gcd &amp; lcm universal property $\,\gcd(a,b)\mid a\mid {\rm lcm}(a,b)\,$ so by transitivity of divisibility ...
Remark $ $  The hint boils down to $\,\inf S \le \sup S\,$ in poset / lattice language. More generally, the hint shows that every common divisor of a set of integers divides every common multiple.
I think the question is asking you to prove that $a\times b$ divides $\mathrm{lcm}(a,b)$. That is, prove that $\mathrm{lcm}(a,b)\times n=a\times b$ where $n\in\mathbb{Z}$.