Let $a, b, c \in \mathbb{N}$. Suppose $\gcd(a, b) = 1$. We prove that $\gcd(ac, b) = \gcd(b, c)$. Assume $\gcd(a, b) = 1$ for all three parts
(a) Let $d \in \mathbb{N}$ be such that $d\mid b$. Prove that $\gcd(a, d) = 1$.
(b) Prove that $d\mid (ac)$ and $d\mid b$ if and only if $d\mid c$ and $d\mid b$. and Euclid’s Lemma. )
(c) Derive from part (b) that $\gcd(ac, b) = \gcd(b, c)$.
So for the part (a) I can use the contradiction: suppose $\gcd(a,b) = r \gt 1$ then $r\mid a$ and $r\mid d$ ... for part (c) that I can use Euclid’s Lemma right?
Any hints for those questions? I got stuck here.