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. 

since $a \,|\, (b+c)$ thus there is $n$ such that $an=b+c$ suppose $\gcd(a,b) = d$ we show that $d=1$.$d \,|\, b$ , $d \,|\, a$ thus $d \,|\, an$ thus $d \,|\, c=an-b$ hence $d \,|\, \gcd(b,c)=1$ so $d=1$
