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

Two consecutive integers are coprime

Twitter proof

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 \(d = 1\).

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