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<p>You are given a positive integer <code>n</code>.</p>
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<p>Continuously replace <code>n</code> with the sum of its <strong>prime factors</strong>.</p>
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<ul>
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<li>Note that if a prime factor divides <code>n</code> multiple times, it should be included in the sum as many times as it divides <code>n</code>.</li>
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</ul>
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<p>Return <em>the smallest value </em><code>n</code><em> will take on.</em></p>
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<p> </p>
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<p><strong class="example">Example 1:</strong></p>
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<pre>
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<strong>Input:</strong> n = 15
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<strong>Output:</strong> 5
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<strong>Explanation:</strong> Initially, n = 15.
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15 = 3 * 5, so replace n with 3 + 5 = 8.
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8 = 2 * 2 * 2, so replace n with 2 + 2 + 2 = 6.
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6 = 2 * 3, so replace n with 2 + 3 = 5.
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5 is the smallest value n will take on.
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</pre>
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<p><strong class="example">Example 2:</strong></p>
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<pre>
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<strong>Input:</strong> n = 3
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<strong>Output:</strong> 3
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<strong>Explanation:</strong> Initially, n = 3.
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3 is the smallest value n will take on.
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</pre>
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<p> </p>
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<p><strong>Constraints:</strong></p>
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<ul>
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<li><code>2 <= n <= 10<sup>5</sup></code></li>
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</ul>
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<p>You are given an integer <code>n</code>. There is a <strong>complete binary tree</strong> with <code>2<sup>n</sup> - 1</code> nodes. The root of that tree is the node with the value <code>1</code>, and every node with a value <code>val</code> in the range <code>[1, 2<sup>n - 1</sup> - 1]</code> has two children where:</p>
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<ul>
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<li>The left node has the value <code>2 * val</code>, and</li>
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<li>The right node has the value <code>2 * val + 1</code>.</li>
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</ul>
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<p>You are also given a 2D integer array <code>queries</code> of length <code>m</code>, where <code>queries[i] = [a<sub>i</sub>, b<sub>i</sub>]</code>. For each query, solve the following problem:</p>
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<ol>
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<li>Add an edge between the nodes with values <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code>.</li>
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<li>Find the length of the cycle in the graph.</li>
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<li>Remove the added edge between nodes with values <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code>.</li>
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</ol>
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<p><strong>Note</strong> that:</p>
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<ul>
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<li>A <strong>cycle</strong> is a path that starts and ends at the same node, and each edge in the path is visited only once.</li>
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<li>The length of a cycle is the number of edges visited in the cycle.</li>
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<li>There could be multiple edges between two nodes in the tree after adding the edge of the query.</li>
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</ul>
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<p>Return <em>an array </em><code>answer</code><em> of length </em><code>m</code><em> where</em> <code>answer[i]</code> <em>is the answer to the</em> <code>i<sup>th</sup></code> <em>query.</em></p>
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<p> </p>
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<p><strong class="example">Example 1:</strong></p>
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<img alt="" src="https://assets.leetcode.com/uploads/2022/10/25/bexample1.png" style="width: 647px; height: 128px;" />
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<pre>
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<strong>Input:</strong> n = 3, queries = [[5,3],[4,7],[2,3]]
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<strong>Output:</strong> [4,5,3]
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<strong>Explanation:</strong> The diagrams above show the tree of 2<sup>3</sup> - 1 nodes. Nodes colored in red describe the nodes in the cycle after adding the edge.
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- After adding the edge between nodes 3 and 5, the graph contains a cycle of nodes [5,2,1,3]. Thus answer to the first query is 4. We delete the added edge and process the next query.
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- After adding the edge between nodes 4 and 7, the graph contains a cycle of nodes [4,2,1,3,7]. Thus answer to the second query is 5. We delete the added edge and process the next query.
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- After adding the edge between nodes 2 and 3, the graph contains a cycle of nodes [2,1,3]. Thus answer to the third query is 3. We delete the added edge.
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</pre>
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<p><strong class="example">Example 2:</strong></p>
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<img alt="" src="https://assets.leetcode.com/uploads/2022/10/25/aexample2.png" style="width: 146px; height: 71px;" />
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<pre>
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<strong>Input:</strong> n = 2, queries = [[1,2]]
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<strong>Output:</strong> [2]
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<strong>Explanation:</strong> The diagram above shows the tree of 2<sup>2</sup> - 1 nodes. Nodes colored in red describe the nodes in the cycle after adding the edge.
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- After adding the edge between nodes 1 and 2, the graph contains a cycle of nodes [2,1]. Thus answer for the first query is 2. We delete the added edge.
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</pre>
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<p> </p>
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<p><strong>Constraints:</strong></p>
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<ul>
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<li><code>2 <= n <= 30</code></li>
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<li><code>m == queries.length</code></li>
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<li><code>1 <= m <= 10<sup>5</sup></code></li>
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<li><code>queries[i].length == 2</code></li>
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<li><code>1 <= a<sub>i</sub>, b<sub>i</sub> <= 2<sup>n</sup> - 1</code></li>
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<li><code>a<sub>i</sub> != b<sub>i</sub></code></li>
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</ul>
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<p>There is an <strong>undirected</strong> graph consisting of <code>n</code> nodes numbered from <code>1</code> to <code>n</code>. You are given the integer <code>n</code> and a <strong>2D</strong> array <code>edges</code> where <code>edges[i] = [a<sub>i</sub>, b<sub>i</sub>]</code> indicates that there is an edge between nodes <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code>. The graph can be disconnected.</p>
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<p>You can add <strong>at most</strong> two additional edges (possibly none) to this graph so that there are no repeated edges and no self-loops.</p>
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<p>Return <code>true</code><em> if it is possible to make the degree of each node in the graph even, otherwise return </em><code>false</code><em>.</em></p>
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<p>The degree of a node is the number of edges connected to it.</p>
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<p> </p>
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<p><strong>Example 1:</strong></p>
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<img alt="" src="https://assets.leetcode.com/uploads/2022/10/26/agraphdrawio.png" style="width: 500px; height: 190px;" />
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<pre>
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<strong>Input:</strong> n = 5, edges = [[1,2],[2,3],[3,4],[4,2],[1,4],[2,5]]
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<strong>Output:</strong> true
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<strong>Explanation:</strong> The above diagram shows a valid way of adding an edge.
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Every node in the resulting graph is connected to an even number of edges.
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</pre>
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<p><strong>Example 2:</strong></p>
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<img alt="" src="https://assets.leetcode.com/uploads/2022/10/26/aagraphdrawio.png" style="width: 400px; height: 120px;" />
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<pre>
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<strong>Input:</strong> n = 4, edges = [[1,2],[3,4]]
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<strong>Output:</strong> true
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<strong>Explanation:</strong> The above diagram shows a valid way of adding two edges.</pre>
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<p><strong>Example 3:</strong></p>
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<img alt="" src="https://assets.leetcode.com/uploads/2022/10/26/aaagraphdrawio.png" style="width: 150px; height: 158px;" />
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<pre>
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<strong>Input:</strong> n = 4, edges = [[1,2],[1,3],[1,4]]
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<strong>Output:</strong> false
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<strong>Explanation:</strong> It is not possible to obtain a valid graph with adding at most 2 edges.</pre>
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<p> </p>
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<p><strong>Constraints:</strong></p>
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<ul>
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<li><code>3 <= n <= 10<sup>5</sup></code></li>
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<li><code>2 <= edges.length <= 10<sup>5</sup></code></li>
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<li><code>edges[i].length == 2</code></li>
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<li><code>1 <= a<sub>i</sub>, b<sub>i</sub> <= n</code></li>
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<li><code>a<sub>i</sub> != b<sub>i</sub></code></li>
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<li>There are no repeated edges.</li>
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</ul>
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<p>You are given a <strong>0-indexed</strong> string array <code>words</code>.</p>
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<p>Two strings are <strong>similar</strong> if they consist of the same characters.</p>
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<ul>
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<li>For example, <code>"abca"</code> and <code>"cba"</code> are similar since both consist of characters <code>'a'</code>, <code>'b'</code>, and <code>'c'</code>.</li>
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<li>However, <code>"abacba"</code> and <code>"bcfd"</code> are not similar since they do not consist of the same characters.</li>
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</ul>
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<p>Return <em>the number of pairs </em><code>(i, j)</code><em> such that </em><code>0 <= i < j <= word.length - 1</code><em> and the two strings </em><code>words[i]</code><em> and </em><code>words[j]</code><em> are similar</em>.</p>
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<p> </p>
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<p><strong class="example">Example 1:</strong></p>
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<pre>
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<strong>Input:</strong> words = ["aba","aabb","abcd","bac","aabc"]
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<strong>Output:</strong> 2
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<strong>Explanation:</strong> There are 2 pairs that satisfy the conditions:
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- i = 0 and j = 1 : both words[0] and words[1] only consist of characters 'a' and 'b'.
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- i = 3 and j = 4 : both words[3] and words[4] only consist of characters 'a', 'b', and 'c'.
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</pre>
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<p><strong class="example">Example 2:</strong></p>
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<pre>
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<strong>Input:</strong> words = ["aabb","ab","ba"]
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<strong>Output:</strong> 3
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<strong>Explanation:</strong> There are 3 pairs that satisfy the conditions:
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- i = 0 and j = 1 : both words[0] and words[1] only consist of characters 'a' and 'b'.
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- i = 0 and j = 2 : both words[0] and words[2] only consist of characters 'a' and 'b'.
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- i = 1 and j = 2 : both words[1] and words[2] only consist of characters 'a' and 'b'.
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</pre>
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<p><strong class="example">Example 3:</strong></p>
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<pre>
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<strong>Input:</strong> words = ["nba","cba","dba"]
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<strong>Output:</strong> 0
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<strong>Explanation:</strong> Since there does not exist any pair that satisfies the conditions, we return 0.</pre>
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<p> </p>
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<p><strong>Constraints:</strong></p>
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<ul>
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<li><code>1 <= words.length <= 100</code></li>
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<li><code>1 <= words[i].length <= 100</code></li>
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<li><code>words[i]</code> consist of only lowercase English letters.</li>
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</ul>
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<p>Given an integer array <code>nums</code>, return <em>all the different possible non-decreasing subsequences of the given array with at least two elements</em>. You may return the answer in <strong>any order</strong>.</p>
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<p> </p>
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<p><strong class="example">Example 1:</strong></p>
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<pre>
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<strong>Input:</strong> nums = [4,6,7,7]
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<strong>Output:</strong> [[4,6],[4,6,7],[4,6,7,7],[4,7],[4,7,7],[6,7],[6,7,7],[7,7]]
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</pre>
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<p><strong class="example">Example 2:</strong></p>
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<pre>
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<strong>Input:</strong> nums = [4,4,3,2,1]
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<strong>Output:</strong> [[4,4]]
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</pre>
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<p> </p>
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<p><strong>Constraints:</strong></p>
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<ul>
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<li><code>1 <= nums.length <= 15</code></li>
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<li><code>-100 <= nums[i] <= 100</code></li>
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</ul>
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