CF1349A.Orac and LCM

For the multiset of positive integers $s=\{s_1,s_2,\dots,s_k\}$ , define the Greatest Common Divisor (GCD) and Least Common Multiple (LCM) of $s$ as follow:

• $\gcd(s)$ is the maximum positive integer $x$ , such that all integers in $s$ are divisible on $x$ .
• $\textrm{lcm}(s)$ is the minimum positive integer $x$ , that divisible on all integers from $s$ .

For example, $\gcd(\{8,12\})=4,\gcd(\{12,18,6\})=6$ and $\textrm{lcm}(\{4,6\})=12$ . Note that for any positive integer $x$ , $\gcd(\{x\})=\textrm{lcm}(\{x\})=x$ .

Orac has a sequence $a$ with length $n$ . He come up with the multiset $t=\{\textrm{lcm}(\{a_i,a_j\})\ |\ i<j\}$ , and asked you to find the value of $\gcd(t)$ for him. In other words, you need to calculate the GCD of LCMs of all pairs of elements in the given sequence.

The first line contains one integer $n\ (2\le n\le 100\,000)$ .

The second line contains $n$ integers, $a_1, a_2, \ldots, a_n$ ( $1 \leq a_i \leq 200\,000$ ).

Print one integer: $\gcd(\{\textrm{lcm}(\{a_i,a_j\})\ |\ i<j\})$ .

For the first example, $t=\{\textrm{lcm}(\{1,1\})\}=\{1\}$ , so $\gcd(t)=1$ .

For the second example, $t=\{120,40,80,120,240,80\}$ , and it's not hard to see that $\gcd(t)=40$ .