Sub-array Sums Divisible By K

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Problem statement

Given an array of integers of size ‘N’ and a positive integer ‘K’. Return the number of non-empty subarrays whose sum is divisible by K.

A subarray is a contiguous subset of an array.


For Example : 
Consider an array of size four. The elements of the array are { -4, 5, 6, 1}. 
The value of K is 4. 
The subarrays whose sum is divisible by 4 are as  follows:
[ -4 ] 
[-4, 5, 6, 1] 
[ 5, 6, 1] 
Hence, there are three subarrays whose sum is divisible by 4.
Detailed explanation ( Input/output format, Notes, Images )
Input Format : 
The first line of input contains an integer T, the number of test cases.

The first line of every test case contains two space-separated integers ‘N’ and ‘K‘ denoting the size of the array and the positive integer K. 

The second line of every test case contains ‘N’ space-separated integers.
Output Format : 
For every test case, print the count of the subarrays whose sum is divisible by K. 

The output of each test case is printed in a separate line.
Note : 
You don’t have to print anything, it has already been taken care of. Just implement the function.
Constraints : 
1 <= T <= 10    
1 <= N <= 10^5
1 <= K <= 10^3
-10^3 <= data <= 10^3

Where ‘data’ denotes the value of the elements of the array.

Time Limit: 1 sec
Sample Input 1 :
2 
5 5 
5 -5 0 -1 2  
1 4
3
Sample Output 1 :
6
0
Explanation of Sample Input 1 :
Test Case 1: Among all the possible subarrays of the given array, there are six subarrays whose sum is divisible by 5. 
[ 5 ] 
[ 5, -5]
[ 5, -5, 0 ]
[ -5, 0 ]
[ -5 ]
[ 0 ] 

Test Case 2: The only subarray [3] is not divisible by 4.
Sample Input 2 :
2
6 5
4 5 0 -2 -3 1
7 3
6 7 1 -2 3 4 9
Sample Output 2 :
7
9
Hint

Can you do this by generating all the subarrays?

Approaches (2)
Brute-Force Approach

The idea is very simple. We will generate all the possible subarrays of the given array. Now, we will find the sum of each subarray and keep track of the count of the subarrays whose sum is divisible by K. 

 

Algorithm

 

  • Initialize a variable ‘result’ that will store the total number of subarrays whose sum is divisible by K.
  • Traverse the array from index 0 to N - 1 where the current index is denoted by i. Initialise a variable ‘sum’ with 0 that will store the sum of the current subarray. 
    We will use another for loop inside this loop, where the current index is denoted by j, to consider all the subarrays starting with the index i and ending at index j.
  • Inside this inner j loop, keep adding the elements to the ‘sum’.
  • Outside the inner j loop, increment the ‘result’ if the sum of the current subarray is divisible by K.
  • Return the ‘result’.
Time Complexity

O(N^2) where ‘N’ is the number of elements in the array.

 

We are using two nested loops to generate all the possible subarrays. This takes O(N^2) time. So, the final time complexity is O(N^2). 

Space Complexity

O(1) 

 

We are not using any extra data structure. Only constant additional space is required.

Code Solution
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Sub-array Sums Divisible By K
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