Is it true that (k,n+k)=d if and only if (k,n)=d?

Question

I solved "Prove that (k,n+k)=1 if and only if (k,n)=1"

but I cannot solve "Is it true that (k,n+k)=d if and only if (k,n)=d?"

I think it is False but I don't know why it is

If m is any integer, then gcd(k,n + mk) = gcd(k, n). How did you solve (k,n)=1? — J. W. Tanner, Apr 01 '19 at 00:57

score 1 · Answer 1 · answered Apr 01 '19 at 02:58

1

Let $d=(k,n)$. Then $d$ divides $k$ and $n,$ so $d$ is a divisor of $n+k$ $(k=da, n=db, n+k=d(b+a)).$

On the other hand, if $e$ divides $k$ and $n+k$ then $e$ divides $n$ ($k=ec, n+k=ef, n=e(f-c)$),

so $e$ divides $d=(k,n)$. Thus, $d=(k,n+k)$.

answered Apr 01 '19 at 02:58

J. W. Tanner

1 Answers1