Show that out of a set of ten consecutive integers, at least one is coprime to all of the others.
Lemma: Out of a set of n consecutive integers, exactly one is divisible by n. (Given).
The Attempt at a Solution
Let a1, a2...a10 be consecutive integers. Let a1≡1(mod10), a2≡ 2(mod10)...a10≡0(mod10).
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 ℤ10, and if so, does this mean that a7 is coprime to the other ai?
Also, sorry that I've been throwing so many questions on here in the last few weeks. Thank you all for being so patient :)