Answer to Question #350954 in Abstract Algebra for Fred

Question

44. Prove that if a and b are different integers, then there exist infinitely 

many positive integers n such that a+n and b+n are relatively prime. 

Answer 1

Let $a$ and $b$ be two different integers. Assume for instance $a<b$, and let $n=(b-a)k+1-a$. For $k$ sufficiently large $n$ will be positive integer. We have $a+n=(b-a)k+1$, $b+n=(b-a)(k+1)+1$, hence $a+n$ and $b+n$ will be positive integers. If we had $d|a+n$ and $d|b+n$, we would have $d|a-b$, and, in view $d|a+n$, also $d|1$, which implies that $d=1$, Thus $(a+n,b+n)=1$.