Problem 1275

8. a) Show that if $a, b$, and $c$ are integers with $(a, b)=(a, c)=1$, then $(a, b c)=1$.
b) Use mathematical induction to show that if $a_{1}, a_{2}, \ldots, a_{n}$ are integers, and $b$ is another integer such that $\left(a_{1}, b\right)=\left(a_{2}, b\right)=\cdots=\left(a_{n}, b\right)=1$, then $\left(a_{1} a_{2} \cdots a_{n}, b\right)=1$