If $\gcd(a,bc)=d$, we can say that $d|a$ and $d|bc$ but what about $d|b$, $d|c$ individually?

Question

I am learning about the proof of the theorem: If $(b,c)=1$, then $(a,b)(a,c)=(a,bc)$. I have completed more than half of the theorem proof but at the end I let $(a,bc)=d$, and i know that $d \mid a,d \mid bc$ can i write also $d \mid c$ and $d \mid b$ individually?. Because $bc$ is the product of two integers. Please clarify my confusion. Thank you.