Modular Product-Sum Ratio
Modular Product-Sum Ratio Let \( f(n) \) be the number of pairs of integers \( (x, y) \) such that \( 1 \le x \le y \le n \) and the product \( xy \) is a multiple of the sum \( x+y \). For example, when \( n = 5 \), there are exactly \( 2 \) valid pairs: \( (2, 2) \), since \( 2 \times 2 = 4 \) is a multiple of \( 2 + 2 = 4 \). \( (4, 4) \), since \( 4 \times 4 = 16 \) is a multiple of \( 4 + 4 = 8 \). Note that \( (3, 6) \) is not valid since \( y \le 5 \) is required. You are also given that: \( f(10) = 6 \) \( f(100) = 110 \) \( f(1000) = 1569 \) Find \( f(10^8) \).