given that n is coprime with 10. prove that there is a number consisting of only 1s such that a is divisible by that number.

Question

anonymous · Answer

So we have $\gcd(n, 10) = 1$ and we want to show some number $11111...|a$?

What is $a$?  What do we know about it?

anonymous · Answer

a|11...1    a is just a random number

anonymous · Answer

But we don't know anything about $a$ yet we are given information about $n$ which is never mentioned later on.

anonymous · Answer

oh sorry, replace a with n

anonymous · Answer

I think there Fermat's little theorem can be used. if we modify it, we can get that every prime number other than 2 and 5 is divisible by a number consisting only of (p-1) 9s.   p|99....9