[ARC122E] Increasing LCMs
Problem Statement
We have a sequence of $N$ positive integers: $A_1,A_2,\cdots,A_N$. You are to rearrange these integers into another sequence $x_1,x_2,\cdots,x_N$, where $x$ must satisfy the following condition:
- Let us define $y_i=\operatorname{LCM}(x_1,x_2,\cdots,x_i)$, where the function $\operatorname{LCM}$ returns the least common multiple of the given integers. Then, $y$ is strictly increasing. In other words, $y_1
Determine whether it is possible to form a sequence $x$ satisfying the condition, and show one such sequence if it is possible.
Constraints
- $1 \leq N \leq 100$
- $2 \leq A_1 < A_2 \cdots < A_N \leq 10^{18}$
- All values in input are integers.
Input
Input is given from Standard Input in the following format:
$N$ $A_1$ $A_2$ $\cdots$ $A_N$
Output
If it is possible to form a sequence $x$ satisfying the condition, print your answer in the following format:
Yes $x_1$ $x_2$ $\cdots$ $x_N$
If it is impossible, print No.
Sample Input 1
3 3 4 6
Sample Output 1
Yes 3 6 4
For $x=(3,6,4)$, we have:
- $y_1=\operatorname{LCM}(3)=3$
- $y_2=\operatorname{LCM}(3,6)=6$
- $y_3=\operatorname{LCM}(3,6,4)=12$
Here, $y_1
Sample Input 2
3 2 3 6
Sample Output 2
No
No permutation of $A$ would satisfy the condition.
Sample Input 3
10 922513 346046618969 3247317977078471 4638516664311857 18332844097865861 81706734998806133 116282391418772039 134115264093375553 156087536381939527 255595307440611247
Sample Output 3
Yes 922513 346046618969 116282391418772039 81706734998806133 255595307440611247 156087536381939527 134115264093375553 18332844097865861 3247317977078471 4638516664311857
巧妙地，考虑倒着构造整个序列。
想一下如何选出一个可以排在最后的数，当且仅当他存在某一个质因子的次数是严格最大的。
可以先用 Pollard-Pho 分解出来判断。
每次选一个可以放在最后的元素，不会使本来可以放的数变成不能放。
但是真的要 Pollard-Pho 吗？
枚举 1 0 6 10^6 106 以内的数进行分解，那么还没分解出来的要不是两个大质数相乘，要不是一个质数。
对于两个大质数的情况，枚举其他的数，取gcd，如果取出来不是 1 我们就分解出来了，否则可以把这个数当成一个数，不影响性质。
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