gcd and lcm - Suppose $b,c in textbf Z^+$ are relatively prime (i.e., $gcd(b,c) = 1$), and $a ,|, (b+c)$. Prove that $gcd(a,b) = 1$ and $gcd(a,c) = 1$

Suppose $b,c \in \textbf Z^+$ are relatively prime (i.e., $\gcd(b,c) = 1$), and $a \,|\, (b+c)$. Prove that $\gcd(a,b) = 1$ and $\gcd(a,c) = 1$.

I've been trying to brainstorm how to prove this. I have determined that $\gcd(b, b + c) = 1$, but I am not sure if this fact will aid in proving this statement at all.