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Set of numbers and function a+b-gcd(a;b)

Source: VII Caucasus Mathematical Olympiad

March 13, 2022

number theoryGCD

Problem Statement

Pete wrote down

21

pairwise distinct positive integers, each not greater than

1,000,000

. For every pair

(a, b)

of numbers written down by Pete, Nick wrote the number

F(a;b)=a+b -\gcd(a;b)

on his piece of paper. Prove that one of Nick’s numbers differs from all of Pete’s numbers.