Show that the following conditions are equivalent:
i) There exist positive integers $a, b$ such that $\gcd(a,b)=d$ and $\operatorname{lcm}(a,b)=m$.
ii) $d\mid m$
[Math] Proof that $\gcd$ divides $\operatorname{lcm}$
divisibilityproof-writing
divisibilityproof-writing
Show that the following conditions are equivalent:
i) There exist positive integers $a, b$ such that $\gcd(a,b)=d$ and $\operatorname{lcm}(a,b)=m$.
ii) $d\mid m$
Best Answer
The one direction is easy since $\gcd(a,b)\mid a$ and $a\mid \operatorname{lcm}(a,b)$.
For the other direction here is a hint.
Assume that $a\mid b$. What are the $\gcd(a,b)$ and $\operatorname{lcm}(a,b)$?