Question

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.

Answer 1

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?

Answer 2

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

Answer 3

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

Answer 4

oh sorry, replace a with n

Answer 5

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