<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




class Solution:

    def get_GCD(self, num1, num2):
        small_num = min(num1, num2)
        big_num = max(num1, num2)

        small_factor = 1
        small_factor_options = []

        while small_num/small_factor >= small_factor:
            big_factor = small_num/small_factor
            
            if big_factor % 1 > 0: # not a whole number
                small_factor += 1
                continue

            small_factor_options.append(small_factor)

            if big_num/big_factor % 1 == 0: # big_factor is factor of small_num and big_num
                return int(big_factor)

            small_factor += 1

        for i in range(len(small_factor_options)-1,-1,-1):
            if small_num % small_factor_options[i] == 0 and big_num % small_factor_options[i] == 0:
                return small_factor_options[i]

        return 1

    def insertGreatestCommonDivisors(self, head: Optional[ListNode]) -> Optional[ListNode]:
        current = head
        while current.next != None:
            gcd = self.get_GCD(current.val, current.next.val)
            gcd_node = ListNode(val=gcd, next = current.next)
            current.next = gcd_node
            current = gcd_node.next


        return head
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