How do I prove that two consecutive integers are relatively prime? In other words, how do I show if two integers have no common factors other than 1? Thanks
Suppose you have some integer whose factorization is
. The consecutive integer is
. Let be a prime dividing . Then doesn't divide otherwise it would divide the difference . This is impossible.
Therefore, there is no prime integer that divide both and . In other words they have no common factors or are relatively prime.