HMMT February

$\frac{1}{a} + \frac{1}{b} = \frac{1}{6} \Rightarrow \frac{a+b}{ab} = \frac{1}{6} \Rightarrow ab = 6a + 6b \Rightarrow ab - 6a - 6b = 0$. Factoring yields $(a-6)(b-6) = 36$. Because $a$ and $b$ are positive integers, only the factor pairs of 36 are possible values of $a-6$ and $b-6$. The possible pairs are: $$\begin{aligned} & a-6=1, b-6=36 \\ & a-6=2, b-6=18 \\ & a-6=3, b-6=12 \\ & a-6=4, b-6=9 \\ & a-6=6, b-6=6 \end{aligned}$$ Because $a \leq b$, the symmetric cases, such as $a-6=12, b-6=3$ are not applicable. Then there are 5 possible pairs.