If $\gcd (a,n) = 1$, how can we prove that $\gcd(a + kn, n) = 1$?

Question

If $\gcd (a,n) = 1$, then how can we prove that $\gcd(a + kn, n) = 1$, where $k$ is any integer?

I feel like I should start with linear combinations theorem, but I'm quite lost. Any help would be greatly appreciated!

$au+nv=1\iff (a+kn)u+n(v-ku)=1$ – zwim Nov 26 '20 at 19:24 — zwim, Nov 26 '20 at 19:24

score 2 · Accepted Answer · answered Nov 26 '20 at 19:15

2

If some prime $p|n$, $p|a+kn$, then $p|kn$ and...

answered Nov 26 '20 at 19:15

Derek Luna

I think the next step would be to prove that p <= 1, though how would that help in the context of the problem? – Mohammed Sarnow Nov 26 '20 at 19:18
Of course, as $p$ is prime, $p \geq 2$. – Derek Luna Nov 26 '20 at 19:19
again, how does that help, sorry for the constant questions, still trying to understand the theory – Mohammed Sarnow Nov 26 '20 at 19:20
We are saying that if there exists a prime $p$ such that $p|n$, and $p|a+kn$, then this prime must also divide $a$. But remember the hypothesis that$ \gcd(a,n)=1$. I am using a proof by contradiction here. – Derek Luna Nov 26 '20 at 19:22
Does that help? – Derek Luna Nov 26 '20 at 19:40
1

sort of, though i am still a bit lost, but it's my own fault, thank you for your help! – Mohammed Sarnow Nov 26 '20 at 20:13

1 Answers1