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Inequality with number of divisors

Source: Iberoamerican MO 2024 Day 1 P1

September 21, 2024
number theoryDivisorsGCD and LCM

Problem Statement

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.
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