Subarray Sums Divisible by K

Subarray Sums Divisible by K

What is this problem about?

"Subarray Sums Divisible by K" is a variation of the "Subarray Sum Equals K" problem. Instead of finding subarrays that sum to a specific value, you need to find the number of contiguous subarrays whose sum is divisible by a given integer k. For example, if the sum is 10 and k is 5, it is a valid subarray. This problem requires a deeper understanding of modular arithmetic alongside prefix sums.

Why is this asked in interviews?

This question is asked by companies like Apple and Citadel because it combines two important concepts: Prefix Sums and Modular Arithmetic properties. Specifically, it uses the property that if two prefix sums have the same remainder when divided by k, then the sum of the subarray between them is divisible by k. It tests if a candidate can apply mathematical properties to optimize a coding solution.

Algorithmic pattern used

The algorithmic pattern used is Prefix Sum with Hashing (Frequency Map) and Modular Arithmetic. You iterate through the array and maintain a running total. For each element, you calculate the remainder of the current sum modulo k. If you have seen the same remainder before, the subarray between the two points has a sum divisible by k. You use a hash map or an array of size k to store the frequencies of these remainders.

Example explanation (use your own example)

Array: [4, 5, 0, -2, -3, 1], k = 5.

1. Remainder count array initialized with {0: 1}.
2. Sum=4, Remainder=4. Count[4] = 1.
3. Sum=9, Remainder=4. We've seen 4 before! Add Count[4] to total result. Count[4] = 2.
4. Sum=9, Remainder=4. Seen 4 again! Add Count[4] (2) to total. Count[4] = 3. The logic continues, tracking how many times each remainder has appeared.

Common mistakes candidates make

A major pitfall is handling negative remainders. In many programming languages, -2 % 5 returns -2 instead of 3. To fix this, you should use (remainder % k + k) % k to ensure the remainder is always positive. Another mistake is the same as in the "Equals K" problem: forgetting to initialize the remainder 0 count to 1. Failing to handle large sums that might overflow an integer type (though not common in Python) is another minor concern.

Interview preparation tip

When preparing for the Subarray Sums Divisible by K interview pattern, ensure you are comfortable with the "Pigeonhole Principle" and its application to remainders. Explain clearly to your interviewer how the remainder logic works: if S1 % K == S2 % K, then (S2 - S1) % K == 0. This mathematical proof is what interviewers look for in a top-tier answer.