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

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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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