Prove that if $d\mid\gcd(a,b)$, then $d\mid a$ and $d\mid b$.

Question

Prove that if $d\mid\gcd(a,b)$, then $d\mid a$ and $d\mid b$. I saw this used in proving another theorem but it was not proved. Does anyone know how to prove it?

Answer 1

Because $\gcd(a, b)$ is the greatest common divisor of $a$ and $b$, it is a common divisor of $a$ and $b$, so we get $\gcd(a, b) \mid a$ and $\gcd(a, b) \mid b.$

Now, if $d \mid \gcd(a, b)$, by the transivity of factors, we get $d \mid a$ and $d \mid b$.

Answer 2

Hint:

$\gcd(a,b) | a $ and $\gcd(a,b) | b$

By transitivity of divisibility, $d | a$ and $d | b$.