<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>

@cache
def pf(n):
    ans = []
    p = 2
    while p * p <= n:
        if n % p == 0:
            ans.append(p)
            while n % p == 0:
                n //= p
        p += 1
    if n > 1:
        ans.append(n)
    return ans

class Solution:
    def gcdSort(self, nums: List[int]) -> bool:
        n = len(nums)
        uf = [-1] * n
        def find(u):
            if uf[u] < 0:
                return u
            uf[u] = find(uf[u])
            return uf[u]
        def unite(u, v):
            u, v = find(u), find(v)
            if u == v:
                return
            if uf[u] < uf[v]:
                u, v = v, u
            uf[v] += uf[u]
            uf[u] = v
        p = {}
        for i, x in enumerate(nums):
            for y in pf(x):
                if y in p:
                    unite(i, p[y])
                p[y] = i
        d = defaultdict(list)
        for i, x in enumerate(nums):
            d[find(i)].append(x)
        for y in d.values():
            y.sort(reverse=True)
        for i in range(n):
            y = d[find(i)]
            nums[i] = y.pop()
        for i in range(n - 1):
            if nums[i + 1] < nums[i]:
                return False
        return True
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