<p>Given the head of a linked list <code>head</code>, in which each node contains an integer value.</p>

<p>Between every pair of adjacent nodes, insert a new node with a value equal to the <strong>greatest common divisor</strong> of them.</p>

<p>Return <em>the linked list after insertion</em>.</p>

<p>The <strong>greatest common divisor</strong> of two numbers is the largest positive integer that evenly divides both numbers.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2023/07/18/ex1_copy.png" style="width: 641px; height: 181px;" />
<pre>
<strong>Input:</strong> head = [18,6,10,3]
<strong>Output:</strong> [18,6,6,2,10,1,3]
<strong>Explanation:</strong> The 1<sup>st</sup> diagram denotes the initial linked list and the 2<sup>nd</sup> diagram denotes the linked list after inserting the new nodes (nodes in blue are the inserted nodes).
- We insert the greatest common divisor of 18 and 6 = 6 between the 1<sup>st</sup> and the 2<sup>nd</sup> nodes.
- We insert the greatest common divisor of 6 and 10 = 2 between the 2<sup>nd</sup> and the 3<sup>rd</sup> nodes.
- We insert the greatest common divisor of 10 and 3 = 1 between the 3<sup>rd</sup> and the 4<sup>th</sup> nodes.
There are no more adjacent nodes, so we return the linked list.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2023/07/18/ex2_copy1.png" style="width: 51px; height: 191px;" />
<pre>
<strong>Input:</strong> head = [7]
<strong>Output:</strong> [7]
<strong>Explanation:</strong> The 1<sup>st</sup> diagram denotes the initial linked list and the 2<sup>nd</sup> diagram denotes the linked list after inserting the new nodes.
There are no pairs of adjacent nodes, so we return the initial linked list.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li>The number of nodes in the list is in the range <code>[1, 5000]</code>.</li>
	<li><code>1 &lt;= Node.val &lt;= 1000</code></li>
</ul>

# Definition for singly-linked list.
# class ListNode:
#     def __init__(self, val=0, next=None):
#         self.val = val
#         self.next = next

def decompose(val, factors):  # 16, {}
    new_fact = None
    if val <= 2:
        if val in factors:
            factors[val]+=1
        else:
            factors[val]=1 #{7:1, 3:1}
    else:
        for i in range(2, val, 1):  #i = 2
            if val%i == 0:
                new_fact = i
                break
        if new_fact is not None:
            if new_fact in factors:
                factors[new_fact]+=1 
            else:
                factors[new_fact] = 1 #factors[7] = 1
            decompose(val//new_fact, factors) #decompose(3, factors)
        else:
            if val in factors:
                factors[val]+=1
            else:
                factors[val]=1 #{7:1, 3:1}

def gcd(val1, val2):  # 21, 24 -> {7:1, 3:1} and {2:3,3:1}
    factors1, factors2 = {},{}
    decompose(val1, factors1)
    decompose(val2, factors2)

    common_fact = 1
    for key in factors1:
        if key in factors2:
            min_count = min(factors1[key], factors2[key])
            common_fact *= key**min_count

    return common_fact


class Solution:
    def insertGreatestCommonDivisors(self, head: Optional[ListNode]) -> Optional[ListNode]:
        test={}
        decompose(16, test)
        print(test)
        if head:
            p0 = head
            p1 = head.next
            while p1 is not None:
                new_p1 = p1.next
                common_fact = gcd(p0.val, p1.val)
                new_node = ListNode(val=common_fact, next=p1)
                p0.next = new_node
                p0 = p1
                p1 = new_p1
            return head
        else:
            return None
        
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