<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>

<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>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img src="https://assets.leetcode.com/uploads/2022/08/13/image-20220813183124-1.png" style="width: 437px; height: 200px;" />
<pre>
<strong>Input:</strong> grid = [[5,2,4],[3,0,5],[0,7,2]], k = 3
<strong>Output:</strong> 2
<strong>Explanation:</strong> There are two paths where the sum of the elements on the path is divisible by k.
The first path highlighted in red has a sum of 5 + 2 + 4 + 5 + 2 = 18 which is divisible by 3.
The second path highlighted in blue has a sum of 5 + 3 + 0 + 5 + 2 = 15 which is divisible by 3.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img src="https://assets.leetcode.com/uploads/2022/08/17/image-20220817112930-3.png" style="height: 85px; width: 132px;" />
<pre>
<strong>Input:</strong> grid = [[0,0]], k = 5
<strong>Output:</strong> 1
<strong>Explanation:</strong> The path highlighted in red has a sum of 0 + 0 = 0 which is divisible by 5.
</pre>

<p><strong class="example">Example 3:</strong></p>
<img src="https://assets.leetcode.com/uploads/2022/08/12/image-20220812224605-3.png" style="width: 257px; height: 200px;" />
<pre>
<strong>Input:</strong> grid = [[7,3,4,9],[2,3,6,2],[2,3,7,0]], k = 1
<strong>Output:</strong> 10
<strong>Explanation:</strong> Every integer is divisible by 1 so the sum of the elements on every possible path is divisible by k.
</pre>

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

<ul>
	<li><code>m == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 5 * 10<sup>4</sup></code></li>
	<li><code>1 &lt;= m * n &lt;= 5 * 10<sup>4</sup></code></li>
	<li><code>0 &lt;= grid[i][j] &lt;= 100</code></li>
	<li><code>1 &lt;= k &lt;= 50</code></li>
</ul>

class Solution:
    def numberOfPaths(self, g: List[List[int]], k: int) -> int:
        def helper(start, north, west, g, k, mod):
            m, n = len(g), len(g[0])
            ## compute north, west updates
            temp = defaultdict(int)
            for key in north[start+1]:
                temp[(key + g[start][start])%k] = north[start+1][key]
                temp[(key + g[start][start])%k] %= mod
            if start != 0:
                for key in west[start+1]:
                    temp[(key + g[start][start])%k] += west[start+1][key]
                    temp[(key + g[start][start])%k] %= mod
            north[start+1] = deepcopy(temp)
            west[start+1] = deepcopy(temp)
            ### 
            for r in range(1, n-start):
                temp = defaultdict(int)
                if start != 0:
                    for key in north[start+1+r]:
                        temp[(key + g[start][start+r])%k] = north[start+1+r][key]
                        temp[(key + g[start][start+r])%k] %= mod
                for key in west[start+1]:
                    temp[(key+g[start][start+r])%k] += west[start+1][key]
                    temp[(key+g[start][start+r])%k] %= mod
                north[start+1+r] = deepcopy(temp)
                west[start+1] = deepcopy(temp)
            ###
            for d in range(1, m-start):
                temp = defaultdict(int)
                for key in north[start+1]:
                    temp[(key + g[start+d][start])%k] = north[start+1][key]
                    temp[(key + g[start+d][start])%k] %= mod
                if start != 0:
                    for key in west[start+1+d]:
                        temp[(key+g[start+d][start])%k] += west[start+1+d][key]
                        temp[(key+g[start+d][start])%k] %= mod
                north[start+1] = deepcopy(temp)
                west[start+1+d] = deepcopy(temp)

            ## traversed all cells?
            if start == min(len(g), len(g[0]))-1:
                return int(north[n][0]) if n > m else int(west[m][0])
            else:
                return helper(start + 1, north, west, g, k, mod)
        
        m, n = len(g), len(g[0])
        north = defaultdict(lambda: {0: 1})
        west = defaultdict(lambda: {0: 1})
        return helper(0, north, west, g, k, 1e9 + 7)
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