Orac and LCM

题意翻译

- 给定长度为 $n$ 的序列 $a_1,a_2,a_3,\ldots,a_n$。 - 设可重集 $S=\{\mathrm{lcm}(a_i,a_j)|1 \leq i<j\leq n\}$。 - 设一个集合的最大公约数为最大的正整数 $d$，满足集合内任意一个元素 $x$ 均有 $d|x$。 - 请你求出 $\gcd(S)$。 - $1 \leq n \leq 10^5$，$1 \leq a_i \leq 2 \times 10^5$。

题目描述

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.

输入输出格式

输入格式

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 $ ).

输出格式

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

输入输出样例

输入样例 #1

2
1 1

输出样例 #1

1

输入样例 #2

4
10 24 40 80

输出样例 #2

40

输入样例 #3

10
540 648 810 648 720 540 594 864 972 648

输出样例 #3

54

说明

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 $ .