Prove that if $c|gcd(a,b)$, then $c|a$ and $c|b$.

Question

Prove that if $c|gcd(a,b)$, then $c|a$ and $c|b$.

I have already been able to prove that if $c|a$ and $c|b$ then $c|gcd(a,b)$. However, I am not sure to prove the converse of this (the question I asked).

Answer 1

If a|b and b|c then a|c.

c|gcd(a,b) and gcd(a,b)|a and gcd(a,b)|b so c|a and c|b.

Answer 2

Hint

If $d=\gcd (a,b)$, then $d\mid a$ and $d\mid b$.

Answer 3

Well if $gcd(a,b) | a$ and $gcd(a,b) | b$, because its a divisor of both. then you have

$$c | gcd(a,b) | a $$ and $$ c | gcd(a,b) | b$$.