abstract algebra - Does $gcd(a,bc)$ divides $gcd(a, b)gcd(a, c)$?

I want to prove that $\gcd(a,bc)$ divides $\gcd(a,b)\gcd(a,c)$ but I can't succeed.

I tried to go with $\gcd(a,b) = sa+tb$ and it didn't work, tried to use the fact that $\gcd(a,b)$ and $\gcd (a,c)$ divide $\gcd (a,bc)$ but got stuck again.
please help.