<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 DSU:
    def __init__(self):
        self.parent = {}
    
    def find(self, p: int):
        if p not in self.parent:
            self.parent[p] = p
        if self.parent[p] != p:
            self.parent[p] = self.find(self.parent[p])
        return self.parent[p]
    
    def join(self, p: int, q: int):
        pp, pq = self.find(p), self.find(q)
        self.parent[pq] = pp
    
MX = int(1e5)
factors = [[] for _ in range(MX+1)]
for x in range(2, MX+1):
    if not factors[x]:
        for y in range(x, MX+1, x):
            factors[y].append(x)

class Solution:
    def gcdSort(self, nums: List[int]) -> bool:
        dsu = DSU()
        fac_seen = {}
        for x in nums:
            for f in factors[x]:
                if f in fac_seen:
                    dsu.join(x, fac_seen[f])
                else:
                    fac_seen[f] = x
                    dsu.find(x)
        grps = defaultdict(list)
        for i, x in enumerate(nums):
            grps[dsu.find(x)].append(x)
            
        for arr in grps.values():
            arr.sort()
        
        pos = defaultdict(int)
        prev = 0
        for x in nums:
            g = dsu.find(x)
            y = grps[g][pos[g]]
            if y < prev:
                return False
            prev = y
            pos[g] += 1
        
        return True
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