Answer to Question #350954 in Abstract Algebra for Fred

Question #350954

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. 


1
Expert's answer
2022-06-16T15:00:46-0400

Let aa and bb be two different integers. Assume for instance a<ba<b, and let n=(ba)k+1an=(b-a)k+1-a. For kk sufficiently large nn will be positive integer. We have a+n=(ba)k+1a+n=(b-a)k+1, b+n=(ba)(k+1)+1b+n=(b-a)(k+1)+1, hence a+na+n and b+nb+n will be positive integers. If we had da+nd|a+n and db+nd|b+n, we would have dabd|a-b, and, in view da+nd|a+n, also d1d|1, which implies that d=1d=1, Thus (a+n,b+n)=1(a+n,b+n)=1.


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