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2016-2017 Fall OMO Problem 10

Source:

November 16, 2016
Online Math Open

Problem Statement

Let a1<a2<a3<a4a_1 < a_2 < a_3 < a_4 be positive integers such that the following conditions hold:
-gcd⁡(ai,aj)>1\gcd(a_i,a_j)>1 holds for all integers 1≤i<j≤41\le i < j\le 4. -gcd⁡(ai,aj,ak)=1\gcd(a_i,a_j,a_k)=1 holds for all integers 1≤i<j<k≤41\le i < j < k\le 4.
Find the smallest possible value of a4a_4.
Proposed by James Lin
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