<p>You are given an integer array <code>nums</code>, and you can perform the following operation <strong>any</strong> number of times on <code>nums</code>:</p>

<ul>
	<li>Swap the positions of two elements <code>nums[i]</code> and <code>nums[j]</code> if <code>gcd(nums[i], nums[j]) &gt; 1</code> where <code>gcd(nums[i], nums[j])</code> is the <strong>greatest common divisor</strong> of <code>nums[i]</code> and <code>nums[j]</code>.</li>
</ul>

<p>Return <code>true</code> <em>if it is possible to sort </em><code>nums</code><em> in <strong>non-decreasing</strong> order using the above swap method, or </em><code>false</code><em> otherwise.</em></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums = [7,21,3]
<strong>Output:</strong> true
<strong>Explanation:</strong> We can sort [7,21,3] by performing the following operations:
- Swap 7 and 21 because gcd(7,21) = 7. nums = [<u><strong>21</strong></u>,<u><strong>7</strong></u>,3]
- Swap 21 and 3 because gcd(21,3) = 3. nums = [<u><strong>3</strong></u>,7,<u><strong>21</strong></u>]
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [5,2,6,2]
<strong>Output:</strong> false
<strong>Explanation:</strong> It is impossible to sort the array because 5 cannot be swapped with any other element.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> nums = [10,5,9,3,15]
<strong>Output:</strong> true
We can sort [10,5,9,3,15] by performing the following operations:
- Swap 10 and 15 because gcd(10,15) = 5. nums = [<u><strong>15</strong></u>,5,9,3,<u><strong>10</strong></u>]
- Swap 15 and 3 because gcd(15,3) = 3. nums = [<u><strong>3</strong></u>,5,9,<u><strong>15</strong></u>,10]
- Swap 10 and 15 because gcd(10,15) = 5. nums = [3,5,9,<u><strong>10</strong></u>,<u><strong>15</strong></u>]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= nums.length &lt;= 3 * 10<sup>4</sup></code></li>
	<li><code>2 &lt;= nums[i] &lt;= 10<sup>5</sup></code></li>
</ul>

class UnionFind:
    def __init__(self, nodes):
        self.rep = {node: node for node in nodes}
        self.rank = {node: 1 for node in nodes}
    def find(self, x):
        root = x
        while root != self.rep[root]:
            root = self.rep[root]
        while x != root:
            temp = self.rep[x]
            self.rep[x] = root
            x = temp
        return root
    def union(self, x, y):
        rep_x = self.find(x)
        rep_y = self.find(y)
        if rep_x == rep_y:
            return
        if self.rank[rep_x] > self.rank[rep_y]:
            self.rep[rep_y] = rep_x
        elif self.rank[rep_x] < self.rank[rep_y]:
            self.rep[rep_x] = rep_y
        else:
            self.rep[rep_y] = rep_x
            self.rank[rep_x] += 1
    def is_connected(self, x, y):
        return self.find(x) == self.find(y)

class Solution:
    def gcdSort(self, nums: List[int]) -> bool:
        def factorize(num): 
            factors = defaultdict(int)
            factor = 2

            while factor*factor <= num:
                while not num%factor:
                    factors[factor] += 1
                    num //= factor

                factor += 1

            if num > 1:
                factors[num] += 1

            return factors

        
        fs = []
        primes = set()
        for num in nums:
            fact = factorize(num)
            fs.append(fact)
            for prime in fact:
                primes.add(prime)

            primes.add(num)
        uf = UnionFind(primes)
        for i, num in enumerate(nums):
            fact = fs[i]

            for prime in fact:
                uf.union(num, prime)
        
        temp = sorted(nums)
        for num, tnum in zip(nums, temp):
            if not uf.is_connected(num, tnum):
                return False
            
        return True
        
2020