<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 Solution:
    def gcdSort(self, nums: List[int]) -> bool:
        @functools.cache
        def getPrimeFactors(num: int) -> Set[int]:
            if num == 1:
                return set()
            if num % 2 == 0:
                s = getPrimeFactors(num // 2)
                return s | {2}
            for i in range(3, int(math.sqrt(num))+1, 2):
                if num % i == 0:
                    s = getPrimeFactors(num // i)
                    return s | {i}
            return {num}
        
        @functools.cache
        def isPrime(p: int) -> bool:
            if p in [2, 3, 5, 7]: return True
            for i in range(3, int(math.sqrt(p)), 2):
                if p % i == 0: return False
            return True
        
        data = {i: i for i in range(2, max(nums)+1)}

        def find(i: int) -> int:
            if i != data[i]:
                data[i] = find(data[i])
            return data[i]
        def union(i: int, j: int):
            pi, pj = find(i), find(j)
            if pi != pj:
                data[pi] = pj
        def connected(i: int, j: int) -> bool:
            return find(i) == find(j)
        
        for num in nums:
            # print(f"{num=}, {getPrimeFactors(num)=}")
            for p in getPrimeFactors(num):
                union(num, p)
        # print(f"{data=}")
        
        numsCopy = [(num, i) for i, num in enumerate(nums)]
        numsCopy.sort()
        # print(f"{nums=}")
        # print(f"{numsCopy=}")
        for j, (num, i) in enumerate(numsCopy):
            if i == j: continue
            if not connected(num, nums[j]):
                # print(f"{i=}, {num=}, {j=}, {nums[j]=}")
                return False
        return True

            
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