TITLE: Prove that every positive integer divides a number such as $70, 700, 7770, 77000$.
QUESTION [23 upvotes]: Prove that every positive integer divides a number such as $70, 700, 7770, 77000$, whose decimal representation consists of one or more $7$’s followed by one or more $0$’s.
Hint:$7$; $77$; $777$; $7777$
I know that I am supposed to use the pigeonhole principle, but I can't figure out how. I know that it probably comes down to proving that all odd numbers divide some number of the form $7, 77, 777$, etc.

REPLY [8 votes]: Take some integer $n$. Consider the numbers 
$$x_i= \underbrace{77...77}_i \text{ mod } n\qquad\text{for $i\in\Bbb N$}$$
The pidgeonhole principle tells you that there are $x_i=x_j$ with $i> j$. Then we have $(x_i-x_j)\equiv 0 \pmod n$ (i.e. $n$ divides $x_i-x_j$), as well as
$$x_i-x_j = \underbrace{77...77}_{i-j}\overbrace{00...00}^{j}.$$