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Part of 2024 Iberoamerican

Problems(1)

Inequality with number of divisors

Source: Iberoamerican MO 2024 Day 1 P1

9/21/2024
For each positive integer nn, let d(n)d(n) be the number of positive integer divisors of nn. Prove that for all pairs of positive integers (a,b)(a,b) we have that: d(a)+d(b)≤d(gcd⁡(a,b))+d(lcm(a,b)) d(a)+d(b) \le d(\gcd(a,b))+d(\text{lcm}(a,b)) and determine all pairs of positive integers (a,b)(a,b) where we have equality case.
number theoryDivisorsGCD and LCM