Let $n$ be a positive integer which is not divisible by $2$ and $5. $ Prove that there is a multiple of $n$ consisting entirely of ones.

Question

Let $n$ be a positive integer which is not divisible by $2$ and $5. $ Prove that there is a multiple of $n$ consisting entirely of ones.

This problem possibly is a duplicate, and I am using the android app in which I don't know how to find existing questions.

I think this can be solved by using Fermat's Little Theorem.

Any help will be appreciated

Answer 1

HINT

Note that $n$ is coprime to $10$.

By Euler's theorem, we have that $$10^{\phi(n)}-1 \equiv 0 \pmod {n}$$

If $n$ is a coprime to $9$, we have that $$\frac{10^{\phi(n)}-1}{9} \equiv 0 \pmod {n}$$ Note $\frac{10^{\phi(n)}-1}{9}$ consisted only of $1$s.