<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:
    M = 100000
    arr = [0]*(M+1)
    for i in range(2, M+1):
        for j in range(i, M+1, i):
            if arr[j] == 0: arr[j] = i

    @lru_cache(None)
    def factorize(self, k):
        primes = []
        while self.arr[k] != 0:
            primes += [self.arr[k]]
            k //= self.arr[k]
        return set(primes)
    
    def gcdSort(self, nums):
        d = defaultdict(set)
        for i, num in enumerate(nums):
            for prime in self.factorize(num):
                d[prime].add(i)

        G = defaultdict(list)
        for x in d:
            for a, b in zip(list(d[x]), list(d[x])[1:]):
                G[a] += [b]
                G[b] += [a]

        visited, ans = set(), {}
        comps, ind, n = defaultdict(list), 0, len(nums)

        def dfs(node, i):
            visited.add(node)
            comps[i].append(node)
            for neib in G[node]:
                if neib in visited: continue
                dfs(neib, i)

        for i in range(n):
            if i not in visited:
                dfs(i, ind)
                ind += 1

        for comp in comps.values():
            for i, l in zip(sorted(comp), sorted(nums[t] for t in comp)):
                ans[i] = l

        return [ans[i] for i in range(n)] == sorted(nums)
128000