Let's prove that if \(k\) is an integer, then \(\gcd(k, k+1) = 1\). That is, any two consecutive integers are coprime.

Let \(k\) be an integer and let \(d\) be the greatest common divisor of \(k\) and \(k + 1\). We have that \((k + 1)/d = k/d + 1/d\) and both \((k + 1)/d\) and \(k/d\) are integers, so \(1/d\) must be an integer and we can only have \(d = 1\).

Twitter proof:

— Mathspp (@mathsppblog) November 14, 2020

Let k be an integer and let d be the greatest common divisor of k and k+1. We have that (k+1)/d=k/d+1/d and both (k+1)/d and k/d are integers, so 1/d must be an integer and we can only have d=1.https://t.co/pItsAnueib