Show that if $a, b$, and $c$ are integers with $(a,b)=(a,c)=1$, then $(a,bc)=1$

Question

Show that if $a, b$, and $c$ are integers with $(a,b)=(a,c)=1$, then $(a,bc)=1$

I don't know exactly that I should use the division algorithm or $(a,b)=d$, $(a/d,b/d)=1$.

This is my first time to study number theory... it's very hard...T^T I can't understand basic thms in this study...

Answer 1

Assume $(a,bc)=d>1$. Then $d$ divides $a$ and $bc$. If $p$ is a prime divisor of $d$ then $p$ divides $a$ and $bc$, hence it divides $b$ or $c$ (definition of being prime). This is absurd because $p$ would divide either $(a,b)$ or $(a,c)$.

Answer 2

Suppose $p$ is a prime number such that $p|(a,bc)$. Then $$p|a\text{ and }p|bc$$ Since $p$ is a prime number, by Euclid's Lemma $p|b$ or $p|c$ (or both). But if $p|b$, then $(a,b)\geq p$, contradiction. The same applies for $c$. Thus, $(a,bc)=1$.