MathDB

Four Numbers with Divisibility Property

Source: Turkey National Mathematical Olympiad 2018

December 2, 2018

number theory

Problem Statement

Let

a_1,a_2,a_3,a_4

be positive integers, with the property that it is impossible to assign them around a circle where all the neighbors are coprime. Let

i,j,k\in\{1,2,3,4\}

with

i \neq j

j\neq k

, and

k\neq i

. Determine the maximum number of triples

(i,j,k)

for which

({\rm gcd}(a_i,a_j))^2|a_k.