Consecutive integers are coprime

Statement

Prove that two consecutive integers $n$ and $n+1$ are always coprime.

Solution

Let $d>0$ be a common divisor of $n$ and $n+1$. Then $d$ also divides their difference: $d \mid (n+1)-n = 1.$ The only positive divisor of $1$ is $1$ itself, so $d=1$. Therefore $\gcd(n,\,n+1) = 1,$ that is, $n$ and $n+1$ are coprime.