## Twitter proof: consecutive integers are coprime

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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 $$d = 1$$.