Problem (Number Theory):
Compute the number of ordered pairs of integers $(a, b)$ such that $1<a \leq 50,1<b \leq 50$, and $\log _{b} a$ is rational.

Solution:
Begin by partitioning $\{2,3, \ldots, 50\}$ into the subsets

$$
\begin{aligned}
A & =\{2,4,8,16,32\} \\
B & =\{3,9,27\} \\
C & =\{5,25\} \\
D & =\{6,36\} \\
E & =\{7,49\} \\
F & =\text { all other integers between } 2 \text { and } 50, \text { inclusive. }
\end{aligned}
$$

If $\log _{b} a$ is rational, then either $a$ and $b$ are both members of one of the sets $A, B, C, D$, or $E$, or $a=b \in F$ (see note below for proof). Then the number of possible ordered pairs is

$$
\begin{aligned}
|A|^{2}+|B|^{2}+|C|^{2}+|D|^{2}+|E|^{2}+|F| & =25+9+4+4+4+35 \\
& =\mathbf{8 1}
\end{aligned}
$$

Final answer: 81