If p is a prime other than $2$ or $5$, show that $p$ divides infinitely many numbers

Question

If $p$ is a prime other than $2$ or $5$, show that $p$ divides infinitely many numbers of the form:

$11, 111, 1111, 11111, 111111, 1111111,\dots $

Hint: Consider the multiplicative order of $10\pmod p$.