prove $a,b,c \in\mathbb{Z}$ Show that $\gcd(a,b)=1$ and $\gcd(a,c)=1$ iff $\gcd(a,[b,c])=1$

Question

I have to prove that if $a,b,c,d \in\mathbb{Z}$ Show that $gcd(a,b)=1$ and $\gcd(a,c)=1$ iff $\gcd(a,[b,c])=1$ I've tried proving the forward direction and im stuck.

Proof: Suppose that $\gcd(a,b)=1$ and $\gcd(a,c)=1$. Let $d=\operatorname{lcm}[b,c]$. By definition we know that $b\mid d$ and $c\mid d$ But by assumption we get $a\nmid b$ and $a \nmid c$... I'm stuck right here. I just need a little hint to show that $a\nmid d$.

Answer 1

Hint:

Suppose, $p\mid(a,[b,c])\implies p\mid a \ \text{and} \ p\mid[b,c]$.

$p\mid[b,c]\implies p\mid b$ or $p\mid c$

....continue!

Prime factorization of $a,b,c$ will be a good way to think about (at least for $\impliedby$ direction).