Solution
Unit Fraction Pairs
Show the problem again
How many ordered pairs $(x, y)$ of positive integers satisfy $\frac{1}{x} + \frac{1}{y} = \frac{1}{12}$? Give an integer.
Worked solution
The answer is 15. Clearing denominators, $12x + 12y = xy$, i.e. $(x - 12)(y - 12) = 144$ (Simon's favourite factoring trick). Both $x, y > 12$ since each unit fraction is less than 1/12, so $x - 12$ and $y - 12$ are positive divisors of 144 with product 144. Every positive divisor $d$ of 144 gives exactly one ordered pair $(12 + d, 12 + 144/d)$. Since $144 = 2^4 \cdot 3^2$ has $(4+1)(2+1) = 15$ divisors, there are 15 ordered pairs (including the symmetric one, $x = y = 24$).
Source: Classic exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.