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

The given problem is to find the number of positive integral solutions of the equation:

\(\frac{1}{x} + \frac{1}{y} = \frac{1}{2025}\)

Let's solve this equation step-by-step:

1. Start by rewriting the equation with a common denominator:
2. Cross-multiply to get rid of the fractions:
3. Rearrange the equation:
4. Add 2025² to both sides to complete the square:
5. Rewrite the left-hand side as a product of terms:
6. Now our task is to find the number of pairs \((a, b)\) such that their product is \(2025^2\). First, factorize 2025:
7. Therefore, \(2025^2\) can be expressed as:
8. In terms of finding pairs \((a, b)\), the number of factors of \(2025^2\) is calculated by adding 1 to each of the exponents in its prime factorization and then taking the product:
9. Thus, there are 45 different pairs \((a, b)\), corresponding to the expressions \((x - 2025)(y - 2025)\) equaling each divisor of \(2025^2\).

Therefore, the number of positive integral solutions for the equation is \(45\).