Beautiful Pairs

Given one or two arrays, compute how many index pairs (or the maximum matching of pairs) satisfy a non‑trivial "beautiful" predicate (e.g., equality/modular/gcd/bitwise/sum constraint) where n is large so brute‑force O(n^2) is infeasible. Expect to use efficient counting and selection techniques—frequency maps, combinatorics, sorting/two‑pointers and sometimes number‑theory or advanced tools (BIT/segment tree/FFT)—to meet tight time/space limits.