Are two integers coprime if they are coprime mod(n)?

1. Aug 10, 2016

jack476

1. The problem statement, all variables and given/known data
Show that out of a set of ten consecutive integers, at least one is coprime to all of the others.

2. Relevant equations
Lemma: Out of a set of n consecutive integers, exactly one is divisible by n. (Given).

3. The attempt at a solution
Let a1, a2...a10 be consecutive integers. Let a11(mod10), a2≡ 2(mod10)...a100(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 :)

2. Aug 10, 2016

jack476

Never mind, 7 has a multiplicative inverse, 3, in the integers modulo 10, so it can divide the other elements.