\problemdata       % Takes 5 *mandatory* arguments
  {modular-pigeons}        % latex-friendly label for the prob.
  {pigeonhole,modular arithmetic} % The topic(s) of the problem content.
  {}                       % Source (if known)
  {S02 q3-2}               % Usage (list ones you are aware of).
  {Nick Matsakis, S02}     % Last revision info (author, date).
\begin{problem}[20]
We want to use the generalized pigeonhole principle to prove the
following statement:
\begin{quote}
Among any set of 150 natural numbers, there must be three numbers, 
$a$, $b$, and $c$, such that all of the pairwise differences, 
$(a-b)$, $(a-c)$, $(b-c)$, are multiples of 70.
\end{quote}

\begin{problemparts}
\problempart[3]
What are the pigeons? 
 \solution[\vspace*{2cm}]{The 150 numbers}

\problempart[3]
What are the holes? 
\solution[\vspace*{2cm}]{the set of numbers $\{ 0, \ldots , 69\}$ }

\problempart[3] How will the pigeons be placed in holes? In other words, what
is the function mapping the pigeons to the holes?

\solution[\vspace*{2cm}]{By taking their value mod 70}

\problempart[11]
Carefully prove the statement using the pigeonhole principle.

\solution[\vspace*{6cm}]{ With the pigeons as the 150 numbers, and the
holes as the set of natural numbers mod 70, we see that there is some
hole with $ \ceil{\frac{150}{70}} = 3$ pigeons, that is, three numbers
$a,b,c$ that are equivalent modulo $70$. Then their differences are
equivalent to $0$ modulo $70$, therefore the differences are all
divisible by 70.}

\end{problemparts}

\end{problem}

