Question

The number of positive integral solutions of $\frac{1}{x} + \frac{1}{y} = \frac{1}{2025}$ is

• A. 105
• B. 45
• C. 135
• D. 25

Hint:

Equations of the form $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ can be transformed into $(x-n)(y-n) = n^2$. The number of positive integer solutions is equal to the number of divisors of $n^2$.

Answer 1

The Correct Option is B

Step 1: Transform the equation.
$\frac{1}{x} + \frac{1}{y} = \frac{1}{2025} \implies \frac{x+y}{xy} = \frac{1}{2025} \implies xy - 2025x - 2025y = 0$

Step 2: Factor using Simon’s Favorite Factoring Trick.
Add $2025^2$ both sides: $xy - 2025x - 2025y + 2025^2 = 2025^2$
Factor left side: $(x-2025)(y-2025) = 2025^2$

Step 3: Count positive integer solutions.
Let $X = x-2025$, $Y = y-2025$. Then $XY = 2025^2$.
Number of positive integer solutions = number of positive divisors of $2025^2$.

Step 4: Factorize 2025.
$2025 = 3^4 \cdot 5^2 \implies 2025^2 = 3^8 \cdot 5^4$

Step 5: Count divisors.
Number of positive divisors = $(8+1)(4+1) = 9 \cdot 5 = 45$