leetcode-problemset/leetcode-cn/originData/paths-in-matrix-whose-sum-is-divisible-by-k.json

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            "questionId": "2521",
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            "categoryTitle": "Algorithms",
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            "title": "Paths in Matrix Whose Sum Is Divisible by K",
            "titleSlug": "paths-in-matrix-whose-sum-is-divisible-by-k",
            "content": "<p>You are given a <strong>0-indexed</strong> <code>m x n</code> integer matrix <code>grid</code> and an integer <code>k</code>. You are currently at position <code>(0, 0)</code> and you want to reach position <code>(m - 1, n - 1)</code> moving only <strong>down</strong> or <strong>right</strong>.</p>\n\n<p>Return<em> the number of paths where the sum of the elements on the path is divisible by </em><code>k</code>. Since the answer may be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>\n\n<p>&nbsp;</p>\n<p><strong class=\"example\">Example 1:</strong></p>\n<img src=\"https://assets.leetcode.com/uploads/2022/08/13/image-20220813183124-1.png\" style=\"width: 437px; height: 200px;\" />\n<pre>\n<strong>Input:</strong> grid = [[5,2,4],[3,0,5],[0,7,2]], k = 3\n<strong>Output:</strong> 2\n<strong>Explanation:</strong> There are two paths where the sum of the elements on the path is divisible by k.\nThe first path highlighted in red has a sum of 5 + 2 + 4 + 5 + 2 = 18 which is divisible by 3.\nThe second path highlighted in blue has a sum of 5 + 3 + 0 + 5 + 2 = 15 which is divisible by 3.\n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n<img src=\"https://assets.leetcode.com/uploads/2022/08/17/image-20220817112930-3.png\" style=\"height: 85px; width: 132px;\" />\n<pre>\n<strong>Input:</strong> grid = [[0,0]], k = 5\n<strong>Output:</strong> 1\n<strong>Explanation:</strong> The path highlighted in red has a sum of 0 + 0 = 0 which is divisible by 5.\n</pre>\n\n<p><strong class=\"example\">Example 3:</strong></p>\n<img src=\"https://assets.leetcode.com/uploads/2022/08/12/image-20220812224605-3.png\" style=\"width: 257px; height: 200px;\" />\n<pre>\n<strong>Input:</strong> grid = [[7,3,4,9],[2,3,6,2],[2,3,7,0]], k = 1\n<strong>Output:</strong> 10\n<strong>Explanation:</strong> Every integer is divisible by 1 so the sum of the elements on every possible path is divisible by k.\n</pre>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>m == grid.length</code></li>\n\t<li><code>n == grid[i].length</code></li>\n\t<li><code>1 &lt;= m, n &lt;= 5 * 10<sup>4</sup></code></li>\n\t<li><code>1 &lt;= m * n &lt;= 5 * 10<sup>4</sup></code></li>\n\t<li><code>0 &lt;= grid[i][j] &lt;= 100</code></li>\n\t<li><code>1 &lt;= k &lt;= 50</code></li>\n</ul>\n",
            "translatedTitle": "矩阵中和能被 K 整除的路径",
            "translatedContent": "<p>给你一个下标从 <strong>0</strong>&nbsp;开始的&nbsp;<code>m x n</code>&nbsp;整数矩阵&nbsp;<code>grid</code>&nbsp;和一个整数&nbsp;<code>k</code>&nbsp;。你从起点&nbsp;<code>(0, 0)</code>&nbsp;出发，每一步只能往 <strong>下</strong>&nbsp;或者往 <strong>右</strong>&nbsp;，你想要到达终点&nbsp;<code>(m - 1, n - 1)</code>&nbsp;。</p>\n\n<p>请你返回路径和能被 <code>k</code>&nbsp;整除的路径数目，由于答案可能很大，返回答案对&nbsp;<code>10<sup>9</sup> + 7</code>&nbsp;<strong>取余</strong>&nbsp;的结果。</p>\n\n<p>&nbsp;</p>\n\n<p><strong>示例 1：</strong></p>\n\n<p><img src=\"https://assets.leetcode.com/uploads/2022/08/13/image-20220813183124-1.png\" style=\"width: 437px; height: 200px;\"></p>\n\n<pre><b>输入：</b>grid = [[5,2,4],[3,0,5],[0,7,2]], k = 3\n<b>输出：</b>2\n<b>解释：</b>有两条路径满足路径上元素的和能被 k 整除。\n第一条路径为上图中用红色标注的路径，和为 5 + 2 + 4 + 5 + 2 = 18 ，能被 3 整除。\n第二条路径为上图中用蓝色标注的路径，和为 5 + 3 + 0 + 5 + 2 = 15 ，能被 3 整除。\n</pre>\n\n<p><strong>示例 2：</strong></p>\n<img src=\"https://assets.leetcode.com/uploads/2022/08/17/image-20220817112930-3.png\" style=\"height: 85px; width: 132px;\">\n<pre><b>输入：</b>grid = [[0,0]], k = 5\n<b>输出：</b>1\n<b>解释：</b>红色标注的路径和为 0 + 0 = 0 ，能被 5 整除。\n</pre>\n\n<p><strong>示例 3：</strong></p>\n<img src=\"https://assets.leetcode.com/uploads/2022/08/12/image-20220812224605-3.png\" style=\"width: 257px; height: 200px;\">\n<pre><b>输入：</b>grid = [[7,3,4,9],[2,3,6,2],[2,3,7,0]], k = 1\n<b>输出：</b>10\n<b>解释：</b>每个数字都能被 1 整除，所以每一条路径的和都能被 k 整除。\n</pre>\n\n<p>&nbsp;</p>\n\n<p><strong>提示：</strong></p>\n\n<ul>\n\t<li><code>m == grid.length</code></li>\n\t<li><code>n == grid[i].length</code></li>\n\t<li><code>1 &lt;= m, n &lt;= 5 * 10<sup>4</sup></code></li>\n\t<li><code>1 &lt;= m * n &lt;= 5 * 10<sup>4</sup></code></li>\n\t<li><code>0 &lt;= grid[i][j] &lt;= 100</code></li>\n\t<li><code>1 &lt;= k &lt;= 50</code></li>\n</ul>\n",
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                "The actual numbers in grid do not matter. What matters are the remainders you get when you divide the numbers by k.",
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