Positive Integer Solutions to Reciprocal Equation

Equation: $\frac{1}{a} + \frac{1}{b} = \frac{1}{9} \implies 9a + 9b = ab \implies ab - 9a - 9b = 0$.

Add 81 to both sides: $(a - 9)(b - 9) = 81$

Since $a \le b$, we look for factor pairs $(x, y)$ of 81 such that $x \le y$ where $x = a-9, y = b-9$:

• $1 \times 81 \implies a - 9 = 1, b - 9 = 81 \implies (10, 90)$
• $3 \times 27 \implies a - 9 = 3, b - 9 = 27 \implies (12, 36)$
• $9 \times 9 \implies a - 9 = 9, b - 9 = 9 \implies (18, 18)$

There are 3 such pairs.