\problemdata                                                             
        {pigeonhole-2} % latex-friendly label for the prob.
        {pigeonhole principle, mods} % The topic(s) of the problem content.  
        {?} % Source (if known)                                          
        {S02 cp7M; S01 TUT7} % Usage (list ones you are aware of). 
        {Tina Wang, S02} % Last revision info (author, date).   
                                                                         
                                                                         

 \begin{problem} 

 Let $S$ be the set of numbers $7, 77, 777, \ldots$. Prove that there
 is an element of $S$ that is divisible by $2001$.

 \solution{We use the pigeonhole principle.
 Let the pigeons be the elements $s \in S$, and
 the holes be the numbers $0 \ldots 2000$.  Let $s$ be placed in
 $s \bmod 2001$.

 Since there are more than 2001 elements of $S$, some hole must have
 two numbers, say $s_i$ and $s_j$, with $i$ and $j$ 7's respectively.
 Without loss of generality, assume that $i<j$.

 Then,
 $s_j - s_i = 77 \ldots 700\ldots0$ where there are $j-i$ 7's and $i$ 0's.
 So, $s_j - s_i = s_{j-i} \cdot 10^i$.
 Since $s_j \bmod 2001 = r = s_i \bmod 2001$,
 $s_j - s_i \bmod 2001 = 0$, so $2001 \mid (s_j - s_i)$.  But
 $gcd(2001,10^i) = 1$, so 2001 must divide $s_{j-i}$.

 }

 \end{problem} 
                                