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

[jack476]

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 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 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 :)

 

[jack476]

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