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## Homework Statement

Show that out of a set of ten consecutive integers, at least one is coprime to all of the others.

## Homework Equations

Lemma: Out of a set of

*n*consecutive integers, exactly one is divisible by

*n*. (Given).

## The Attempt at a Solution

Let

*a*be consecutive integers. Let

_{1}, a_{2}...a_{10}*a*1(mod10),

_{1}≡*a*mod10)...

_{2}≡ 2(*a*0(mod10).

_{10}≡Observe that 7 is coprime to the other integers from 1 to 10. Does this mean that 7(mod10) is coprime to the other integers in ℤ

_{10}, and if so, does this mean that

*a*is coprime to the other

_{7}*a*?

_{i}Also, sorry that I've been throwing so many questions on here in the last few weeks. Thank you all for being so patient :)