Prove that $(m,n) = (s,n) = 1$ if and only if $(ms,n)=1$

Question

About the notation: Denotes $(a,b)$ the greatest common divisor between $a$ and $b$.

Now, about the exercise, what I did was the following:

Since $(m, n) = (s, n) = 1$,

$ma + nb = sc + nd = 1$ then,

$ma + nb-sc-nd = 1$

$ma-sc + (b-d) n = 1$

let $b-d = j$ then the previous equality I can write it like this: $ma-sc + jn = 1$

I want to get to that $r (ms) + jn = 1$ but I can not think of how to continue to do it.

Besides, is it okay what I did until now?

Answer 1

First lets assume $(m,n)=(s,n)=1$ then there are $x_1,x_2,y_1,y_2\in\mathbb{Z}$ such that $mx_1+ny_1=1$ and $sx_2+ny_2=1$ multiplying them we get $msx+ny=1$ for some integers $x,y$.

Now the other way i.e. lets assume $(ms,n)=1$ and let $(m,n)=k$ then $k|m\Rightarrow k|ms$ but $k|n$ thus $k|(ms,n)$ so $k=1$. Similarly we can show that $(s,n)=1$.