Pigeonhole question with finding a number.

Question

Show that there is a number consisting only of 1’s that is divisible by 2001.

I know that it relates to the Quotient-Remainder Theorem and I got m=2001q+r, r: [0,2001). But I don't know how it relates to the Pigeonhole Principle.

Answer 1

Hint: $2001$ is relatively prime to $10$, so you can disregard trailing $0$'s. I.e., $111\ldots11$ is divisible by $2001$ if and only if $111\ldots11000\ldots00$ is.

Answer 2

Hint: Use pigeonhole principle to show that two of such numbers must have the same remainder when divided by $2001$. Thus their difference must be divisible by $2001$. What form is that difference?