<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, n):
        self.parent = list(range(n))
        self.rank = [0] * n
    
    def find(self, x):
        if x!=self.parent[x]:
            self.parent[x] = self.find(self.parent[x])
        return self.parent[x]

    def union(self,x,y):
        px = self.find(x)
        py = self.find(y)
        if px == py:
            return False
        rx = self.rank[px]
        ry = self.rank[py]
        if rx > ry:
            self.parent[py] = px
        elif ry > rx:
            self.parent[px] = py
        else:
            self.parent[px] = py
            self.rank[py] += 1
        return True


class Solution:

    def get_prime(self, n):
        for i in range(2, int(math.sqrt(n))+1):
            if n % i == 0:
                return self.get_prime(n//i) | set([i])
        return set([n])

    

    
    def gcdSort(self, nums: List[int]) -> bool:
        
        n = len(nums)
        p = 0
        prime_map = defaultdict(list)
        uf = UnionFind(n)
        for i, num in enumerate(nums):
            pm = self.get_prime(num)
            for p in pm:
                prime_map[p].append(i)

        for p in prime_map:
            for i in range(len(prime_map[p])-1):
                uf.union(prime_map[p][i], prime_map[p][i+1])

        expected = [(num, i) for i, num in enumerate(nums)]
        expected.sort()
        for i in range(n):
            if nums[i] == expected[i][0] or uf.find(i) == uf.find(expected[i][1]):
                continue
            else:
                return False
        return True



        


        

        
2029