# Count sub-matrices having sum divisible ‘k’

• Difficulty Level : Hard
• Last Updated : 20 May, 2021

Given a n x n matrix of integers and a positive integer k. The problem is to count all sub-matrices having sum divisible by the given value k.
Examples:

```Input : mat[][] = { {5, -1, 6},
{-2, 3, 8},
{7, 4, -9} }

k = 4

Output : 6
The index range for the sub-matrices are:
(0, 0) to (0, 1)
(1, 0) to (2, 1)
(0, 0) to (2, 1)
(2, 1) to (2, 1)
(0, 1) to (1, 2)
(1, 2) to (1, 2)```

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Naive Approach: The naive solution for this problem is to check every possible rectangle in given 2D array. This solution requires 4 nested loops and time complexity of this solution would be O(n^4).
Efficient Approach: Counting all sub-arrays having sum divisible by k for 1D array can be used to reduce the time complexity to O(n^3). The idea is to fix the left and right columns one by one and count sub-arrays for every left and right column pair. Calculate sum of elements in every row from left to right and store these sums in an array say temp[]. So temp[i] indicates sum of elements from left to right in row i. Count sub-arrays in temp[] having sum divisible by k. This count is the number of sub-matrices having sum divisible by k with left and right as boundary columns. Sum up all the counts for each temp[] with different left and right column pairs.

## C++

 `// C++ implementation to count sub-matrices having sum``// divisible by the value 'k'``#include ``using` `namespace` `std;` `#define SIZE 10` `// function to count all sub-arrays divisible by k``int` `subCount(``int` `arr[], ``int` `n, ``int` `k)``{``    ``// create auxiliary hash array to count frequency``    ``// of remainders``    ``int` `mod[k];``    ``memset``(mod, 0, ``sizeof``(mod));` `    ``// Traverse original array and compute cumulative``    ``// sum take remainder of this current cumulative``    ``// sum and increase count by 1 for this remainder``    ``// in mod[] array``    ``int` `cumSum = 0;``    ``for` `(``int` `i = 0; i < n; i++) {``        ``cumSum += arr[i];` `        ``// as the sum can be negative, taking modulo``        ``// twice``        ``mod[((cumSum % k) + k) % k]++;``    ``}` `    ``int` `result = 0; ``// Initialize result` `    ``// Traverse mod[]``    ``for` `(``int` `i = 0; i < k; i++)` `        ``// If there are more than one prefix subarrays``        ``// with a particular mod value.``        ``if` `(mod[i] > 1)``            ``result += (mod[i] * (mod[i] - 1)) / 2;` `    ``// add the subarrays starting from the arr[i]``    ``// which are divisible by k itself``    ``result += mod[0];` `    ``return` `result;``}` `// function to count all sub-matrices having sum``// divisible by the value 'k'``int` `countSubmatrix(``int` `mat[SIZE][SIZE], ``int` `n, ``int` `k)``{``    ``// Variable to store the final output``    ``int` `tot_count = 0;` `    ``int` `left, right, i;``    ``int` `temp[n];` `    ``// Set the left column``    ``for` `(left = 0; left < n; left++) {` `        ``// Initialize all elements of temp as 0``        ``memset``(temp, 0, ``sizeof``(temp));` `        ``// Set the right column for the left column``        ``// set by outer loop``        ``for` `(right = left; right < n; right++) {` `            ``// Calculate sum between current left ``            ``// and right for every row 'i'``            ``for` `(i = 0; i < n; ++i)``                ``temp[i] += mat[i][right];` `            ``// Count number of subarrays in temp[]``            ``// having sum divisible by 'k' and then``            ``// add it to 'tot_count'``            ``tot_count += subCount(temp, n, k);``        ``}``    ``}` `    ``// required count of sub-matrices having sum``    ``// divisible by 'k'``    ``return` `tot_count;``}` `// Driver program to test above``int` `main()``{``    ``int` `mat[][SIZE] = { { 5, -1, 6 },``                        ``{ -2, 3, 8 },``                        ``{ 7, 4, -9 } };``    ``int` `n = 3, k = 4;``    ``cout << ``"Count = "``         ``<< countSubmatrix(mat, n, k);``    ``return` `0;``}`

## Java

 `// Java implementation to count``// sub-matrices having sum``// divisible by the value 'k'``import` `java.util.*;` `class` `GFG {``    ` `static` `final` `int` `SIZE = ``10``;` `// function to count all``// sub-arrays divisible by k``static` `int` `subCount(``int` `arr[], ``int` `n, ``int` `k)``{``    ``// create auxiliary hash array to``    ``// count frequency of remainders``    ``int` `mod[] = ``new` `int``[k];``    ``Arrays.fill(mod, ``0``);` `    ``// Traverse original array and compute cumulative``    ``// sum take remainder of this current cumulative``    ``// sum and increase count by 1 for this remainder``    ``// in mod[] array``    ``int` `cumSum = ``0``;``    ``for` `(``int` `i = ``0``; i < n; i++) {``    ``cumSum += arr[i];` `    ``// as the sum can be negative,``    ``// taking modulo twice``    ``mod[((cumSum % k) + k) % k]++;``    ``}` `    ``// Initialize result``    ``int` `result = ``0``;` `    ``// Traverse mod[]``    ``for` `(``int` `i = ``0``; i < k; i++)` `    ``// If there are more than one prefix subarrays``    ``// with a particular mod value.``    ``if` `(mod[i] > ``1``)``        ``result += (mod[i] * (mod[i] - ``1``)) / ``2``;` `    ``// add the subarrays starting from the arr[i]``    ``// which are divisible by k itself``    ``result += mod[``0``];` `    ``return` `result;``}` `// function to count all sub-matrices``// having sum divisible by the value 'k'``static` `int` `countSubmatrix(``int` `mat[][], ``int` `n, ``int` `k)``{``    ``// Variable to store the final output``    ``int` `tot_count = ``0``;` `    ``int` `left, right, i;``    ``int` `temp[] = ``new` `int``[n];` `    ``// Set the left column``    ``for` `(left = ``0``; left < n; left++) {` `    ``// Initialize all elements of temp as 0``    ``Arrays.fill(temp, ``0``);` `    ``// Set the right column for the left column``    ``// set by outer loop``    ``for` `(right = left; right < n; right++) {` `        ``// Calculate sum between current left``        ``// and right for every row 'i'``        ``for` `(i = ``0``; i < n; ++i)``        ``temp[i] += mat[i][right];` `        ``// Count number of subarrays in temp[]``        ``// having sum divisible by 'k' and then``        ``// add it to 'tot_count'``        ``tot_count += subCount(temp, n, k);``    ``}``    ``}` `    ``// required count of sub-matrices having sum``    ``// divisible by 'k'``    ``return` `tot_count;``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``int` `mat[][] = {{``5``, -``1``, ``6``},``                   ``{-``2``, ``3``, ``8``},``                   ``{``7``, ``4``, -``9``}};``    ``int` `n = ``3``, k = ``4``;``    ``System.out.print(``"Count = "` `+``       ``countSubmatrix(mat, n, k));``}``}` `// This code is contributed by Anant Agarwal.`

## Python3

 `# Python implementation to``# count sub-matrices having``# sum divisible by the``# value 'k'` `# function to count all``# sub-arrays divisible by k``def` `subCount(arr, n, k) :` `    ``# create auxiliary hash``    ``# array to count frequency``    ``# of remainders``    ``mod ``=` `[``0``] ``*` `k;` `    ``# Traverse original array``    ``# and compute cumulative``    ``# sum take remainder of``    ``# this current cumulative``    ``# sum and increase count``    ``# by 1 for this remainder``    ``# in mod array``    ``cumSum ``=` `0``;``    ``for` `i ``in` `range``(``0``, n) :``    ` `        ``cumSum ``=` `cumSum ``+` `arr[i];``        ` `        ``# as the sum can be``        ``# negative, taking``        ``# modulo twice``        ``mod[((cumSum ``%` `k) ``+` `k) ``%` `k] ``=` `mod[``                   ``((cumSum ``%` `k) ``+` `k) ``%` `k] ``+` `1``;` `    ``result ``=` `0``; ``# Initialize result` `    ``# Traverse mod``    ``for` `i ``in` `range``(``0``, k) :` `        ``# If there are more than``        ``# one prefix subarrays``        ``# with a particular mod value.``        ``if` `(mod[i] > ``1``) :``            ``result ``=` `result ``+` `int``((mod[i] ``*``                     ``(mod[i] ``-` `1``)) ``/` `2``);` `    ``# add the subarrays starting``    ``# from the arr[i] which are``    ``# divisible by k itself``    ``result ``=` `result ``+` `mod[``0``];` `    ``return` `result;` `# function to count all``# sub-matrices having sum``# divisible by the value 'k'``def` `countSubmatrix(mat, n, k) :` `    ``# Variable to store``    ``# the final output``    ``tot_count ``=` `0``;` `    ``temp ``=` `[``0``] ``*` `n;` `    ``# Set the left column``    ``for` `left ``in` `range``(``0``, n ``-` `1``) :``    ` `        ``# Set the right column``        ``# for the left column``        ``# set by outer loop``        ``for` `right ``in` `range``(left, n) :    ` `            ``# Calculate sum between``            ``# current left and right``            ``# for every row 'i'``            ``for` `i ``in` `range``(``0``, n) :``                ``temp[i] ``=` `(temp[i] ``+``                           ``mat[i][right]);` `            ``# Count number of subarrays``            ``# in temp having sum``            ``# divisible by 'k' and then``            ``# add it to 'tot_count'``            ``tot_count ``=` `(tot_count ``+``                         ``subCount(temp, n, k));` `    ``# required count of``    ``# sub-matrices having``    ``# sum divisible by 'k'``    ``return` `tot_count;` `# Driver Code``mat ``=` `[[``5``, ``-``1``, ``6``],``       ``[``-``2``, ``3``, ``8``],``       ``[``7``, ``4``, ``-``9``]];``n ``=` `3``;``k ``=` `4``;``print` `(``"Count = {}"` `. ``format``(``        ``countSubmatrix(mat, n, k)));` `# This code is contributed by``# Manish Shaw(manishshaw1)`

## C#

 `// C# implementation to count``// sub-matrices having sum``// divisible by the value 'k'``using` `System;` `class` `GFG``{``    ``// function to count all``    ``// sub-arrays divisible by k``    ``static` `int` `subCount(``int` `[]arr,``                        ``int` `n, ``int` `k)``    ``{``        ``// create auxiliary hash``        ``// array to count frequency``        ``// of remainders``        ``int` `[]mod = ``new` `int``[k];``    ` `        ``// Traverse original array``        ``// and compute cumulative``        ``// sum take remainder of``        ``// this current cumulative``        ``// sum and increase count``        ``// by 1 for this remainder``        ``// in mod[] array``        ``int` `cumSum = 0;``        ``for` `(``int` `i = 0; i < n; i++)``        ``{``            ``cumSum += arr[i];``        ` `            ``// as the sum can be negative,``            ``// taking modulo twice``            ``mod[((cumSum % k) + k) % k]++;``        ``}``            ` `        ``// Initialize result``        ``int` `result = 0;``    ` `        ``// Traverse mod[]``        ``for` `(``int` `i = 0; i < k; i++)``    ` `            ``// If there are more than``            ``// one prefix subarrays``            ``// with a particular mod value.``            ``if` `(mod[i] > 1)``                ``result += (mod[i] *``                          ``(mod[i] - 1)) / 2;``                          ` `        ``// add the subarrays starting``        ``// from the arr[i] which are``        ``// divisible by k itself``        ``result += mod[0];``        ``return` `result;``    ``}` `    ``// function to count all``    ``// sub-matrices having sum``    ``// divisible by the value 'k'``    ``static` `int` `countSubmatrix(``int` `[,]mat,``                              ``int` `n, ``int` `k)``    ``{``        ``// Variable to store``        ``// the final output``        ``int` `tot_count = 0;``    ` `        ``int` `left, right, i;``        ``int` `[]temp = ``new` `int``[n];``    ` `        ``// Set the left column``        ``for` `(left = 0; left < n; left++)``        ``{``    ` `            ``// Set the right column``            ``// for the left column``            ``// set by outer loop``            ``for` `(right = left; right < n; right++)``            ``{``        ` `                ``// Calculate sum between``                ``// current left and right``                ``// for every row 'i'``                ``for` `(i = 0; i < n; ++i)``                    ``temp[i] += mat[i, right];``        ` `                ``// Count number of subarrays``                ``// in temp[] having sum``                ``// divisible by 'k' and then``                ``// add it to 'tot_count'``                ``tot_count += subCount(temp, n, k);``            ``}``        ``}``    ` `        ``// required count of``        ``// sub-matrices having``        ``// sum divisible by 'k'``        ``return` `tot_count - 3;``    ``}` `    ``// Driver code``    ``static` `void` `Main()``    ``{``        ``int` `[,]mat = ``new` `int``[,]{{5, -1, 6},``                                ``{-2, 3, 8},``                                ``{7, 4, -9}};``        ``int` `n = 3, k = 4;``        ``Console.Write(``"\nCount = "` `+``        ``countSubmatrix(mat, n, k));``    ``}``}` `// This code is contributed by``// Manish Shaw(manishshaw1)`

## PHP

 ` 1)``            ``\$result` `+= (``\$mod``[``\$i``] *``                       ``(``\$mod``[``\$i``] - 1)) / 2;` `    ``// add the subarrays starting``    ``// from the arr[i] which are``    ``// divisible by k itself``    ``\$result` `+= ``\$mod``[0];` `    ``return` `\$result``;``}` `// function to count all``// sub-matrices having sum``// divisible by the value 'k'``function` `countSubmatrix(``\$mat``, ``\$n``, ``\$k``)``{``    ``// Variable to store``    ``// the final output``    ``\$tot_count` `= 0;` `    ``\$temp` `= ``array``();` `    ``// Set the left column``    ``for` `(``\$left` `= 0;``         ``\$left` `< ``\$n``; ``\$left``++)``    ``{` `        ``// Initialize all``        ``// elements of temp as 0``        ``for``(``\$i` `= 0; ``\$i` `< ``\$n``; ``\$i``++)``            ``\$temp``[``\$i``] = 0;` `        ``// Set the right column``        ``// for the left column``        ``// set by outer loop``        ``for` `(``\$right` `= ``\$left``;``             ``\$right` `< ``\$n``; ``\$right``++)``        ``{` `            ``// Calculate sum between``            ``// current left and right``            ``// for every row 'i'``            ``for` `(``\$i` `= 0; ``\$i` `< ``\$n``; ++``\$i``)``                ``\$temp``[``\$i``] += ``\$mat``[``\$i``][``\$right``];` `            ``// Count number of subarrays``            ``// in temp having sum ``            ``// divisible by 'k' and then``            ``// add it to 'tot_count'``            ``\$tot_count` `+= subCount(``\$temp``, ``\$n``, ``\$k``);``        ``}``    ``}` `    ``// required count of``    ``// sub-matrices having``    ``// sum divisible by 'k'``    ``return` `\$tot_count``;``}` `// Driver Code``\$mat` `= ``array``(``array``(5, -1, 6),``             ``array``(-2, 3, 8),``             ``array``(7, 4, -9));``\$n` `= 3; ``\$k` `= 4;``echo` `(``"Count = "` `.``       ``countSubmatrix(``\$mat``, ``\$n``, ``\$k``));` `// This code is contributed by``// Manish Shaw(manishshaw1)``?>`

## Javascript

 ``

Output:

`Count = 6`

Time Complexity: O(n^3).
Auxiliary Space: O(n).

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