TITLE: A combinatorial number theory question (pigeonhole principle)
QUESTION [5 upvotes]: Let $n$ be a positive integer such that $n$ and $10$ are coprime. Prove that $n|11\cdots11$ for some $11\cdots11$ in base 10 representation.
This problem is about pigeonhole principle, I have a great difficulty in setting up the pigeonhole. Is there any general rule of thumbs in setting up them? thank you.

REPLY [6 votes]: Whenever you see such questions, try to find the pigeons first!
Hint: Consider the sequence $1,11,111,1111,...,\overbrace{11\cdots 11}^{n+1 \;\text{times}}$ as pigeons. 
Can you conclude from here? 
Remark: Let $n \in \mathbb{N}$. Given $n+1$ integers $a_1,\cdots,a_{n+1}$ there exist $i\neq j,$ s.t. $a_i \equiv a_j \;(\text{mod}\; n).$
The key is to write each number $a_k=nq_k+r_k$ where $0\leq r_k \leq n-1$ and observe that since each $r_k$ varies from $0$ to $n-1$ and there are $n$ possibilities, by taking $0,1,\cdots,n-1$ as pigeonholes and $r_k$s as pigeons, one would conclude that at least two pigeons (i.e. at least two $a_k$s) share the same pigeonholes (i.e. have the same remainder).
Possible next exercise: 

Show that there is a power of $n$ which terminates in $0001$ in base $10.$