How to prove the existence of a number of the form 111...111 that can be divided by a prime?

Question

Let $p$ be a prime number ($ p \ne 2, p \ne 5$). How can I prove that this prime number divides at least one number with a decimal representation that consists only of $1$s ($111$, $1111$, etc.)

Answer 1

Hint 1: Fermats Little Theorem says fo all $a$ with $\gcd (a,p) =1$ and $p $ prime, that $a^{p-1}\equiv 1\mod p $.

Hint 2: $10=2*5$ and $p\ne 2; p\ne 5$.

Hint 3: $10^k - 1$ is $99999...99999$.

Hint 4: $p$ may or may not be equal to $3$.