Chef and GCD Game

Chef has a sequence $A$ consisting of $N$ positive integers. He is playing a game with this sequence. The goal of the game is to maximize the greatest common divisor of all elements of the sequence. Chef may perform the following operation an arbitrary number of times (including zero): - Choose a positive integer $d$ and a valid index $i$ such that $A_i$ is divisible by $d^2$. - Divide $A_i$ by