\documentstyle[12pt,twoside,amssymb]{article}
\input{macros-course}
\begin{document}
\begin{problems}

% This problem is a little hard, maybe as a bonus problem?
% Alternatively, we can make this a multiple-part problem leading up
% to the proof.
\problem
Let $m$ be any non-negative integer, written in the standard decimal
notation:
\[
m  = d_kd_{k-1}\ldots d_1d_0
\]
where $0 \leq d_i \leq 9$ for all i. Define a function $T : \naturals
\rightarrow \naturals$ by $T(m) = (d_k+1)(d_{k-1}+1) \cdots (d_1+1)(d_0 +
1)$, using the decimal expansion of $m$ as above.

Prove that
\[
\sum_{n=0}^{10^k - 1} T(n) = 55^k
\]

% Some easy facts about GCD's
\problem
\ppart Show that $\gcd(a,b) = \gcd(b,a-b)$.
\ppart Show that if $a = 2^e x$ and $b = 2^f y$ where $2$ does not divide
$x$ nor $y$, then $\gcd(a,b) = 2^{\min(e,f)}\gcd(a,b)$.
\ppart Solve the equation $40x + 52y = 18$


% True/False questions
% Draw up a Hasse diagram and make some claims about schedules,
% sizes of chains, antichains, greatest elements, least elements, etc.

\end{problems}
\end{document}
