A combinatorial number theory question (pigeonhole principle) 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.

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 $10.$

