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

Question

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).

"Inverse" is the wrong word. The question in the title is the converse of the thing you've already proved. — Jack Lee, Oct 14 '15 at 21:35

score 1 · Answer 1 · answered Oct 14 '15 at 21:35

1

Hint

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

answered Oct 14 '15 at 21:35

Surb

55,662

score 0 · Answer 2 · answered Oct 14 '15 at 21:35

0

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

answered Oct 14 '15 at 21:35

fleablood

124,253

score 0 · Answer 3 · answered Oct 14 '15 at 21:35

0

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

answered Oct 14 '15 at 21:35

Stefan Hante

2,616

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

3 Answers3