(a) Each of Peter and Basil thinks of three positive integers. For each pair of his numbers, Peter writes down the greatest common divisor of the two numbers. For each pair of his numbers, Basil writes down the least common multiple of the two numbers. If both Peter and Basil write down the same three numbers, prove that these three numbers are equal to each other. (b) Can the analogous result be proved if each of Peter and Basil thinks of four positive integers instead?
Problem
Source: Tournament of Towns 2007 - Fall - Junior A-Level - P2
Tags: greatest common divisor, least common multiple, combinatorics unsolved, combinatorics