Prove that n and (n+1) must be relatively prime?

Question

how do i go about proving the following:
let n be a positive integer. prove that n and (n+1) must be relatively prime. help!

Answer 1

Assume that n and n+1 have a common integer factor x

so n/x and (n+1)/x are both integers.

Now (n+1)/x = n/x + 1/x

We know that n/x is an integer (from the above assumption), so this also means that 1/x is an integer.

The only integer value x such that 1/x is an integer is 1.

Therefore the only common factor that n and n+1 have is 1; thus they are relatively prime.

Answer 2

Two integers a and b are relatively prime if and only if there are integers x and y such that ax + by = 1. For n and n + 1, this is easy: (n + 1)*(1) + n*(-1) = 1.