10 Consecutive Integers contain Coprime Integer

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Theorem

Let $n \in \Z$ be an integer.

Let $S := \left\{ {n, n + 1, n + 2, \ldots, n + 9}\right\}$ be the set of $10$ consecutive integers starting from $n$.


Then at least one element of $S$ is coprime to every other element of $S$.


Proof


Historical Note

According to 1997: David Wells: Curious and Interesting Numbers (2nd ed.), this result is the work of B.G. Eke, but no confirmation of this fact, or any biographical details, have yet been found to corroborate this.


Sources