Prove if the remainder is relatively prime to the divisor, the dividend is also relatively prime to the divisor and vice versa?

Question

The remainder of $a÷b$ is $c$, both $a$ and $b$ are positive integers.

How to prove:

1. $gcd(c,b)=1 \iff gcd(a,b)=1$

2. $gcd(a,b)=1 \iff gcd(c,b)=1$

Answer 1

let $a = bq + c$ for some integer $q$.

Then if say $gcd(a,b) = 1$, and $gcd(b,c) = m$,

then, from $a = bq +c$ ;

we will get $m|a$ but $m$ also divides $b$ $\implies$ $m$ is a common divisor of $a$ and $b$ , And the only common divisor of $a$ and $b$ is 1.