Continuous Subarray Sum

Continuous Subarray Sum

What is this problem about?

The Continuous Subarray Sum interview question asks you to determine if an integer array contains a continuous subarray of size at least two whose elements sum up to a multiple of a given integer k. A multiple of k is any integer n * k, including 0 if k != 0.

Why is this asked in interviews?

Companies like Meta, Amazon, and Bloomberg use the Continuous Subarray Sum coding problem to test a candidate's mastery of the Remainder Theorem combined with the Prefix Sum technique. It requires moving beyond simple sums to modular arithmetic, which is a common pattern in high-scale data processing and cryptography.

Algorithmic pattern used

This utilizes the Array, Math, Hash Table, Prefix Sum interview pattern.

• If the running sum up to index i has a remainder r when divided by k, and the running sum up to index j also has the same remainder r, then the sum of the elements between i and j must be a multiple of k.
• We use a Hash Map to store the first occurrence of each remainder.
• Key: Remainder (prefix_sum % k).
• Value: The index where this remainder first appeared.

Example explanation

Array: [23, 2, 4, 6, 7], k = 6

1. Map: {0: -1} (Initial state).
2. Index 0: Sum=23, 23 % 6 = 5. Map: {0: -1, 5: 0}.
3. Index 1: Sum=25, 25 % 6 = 1. Map: {0: -1, 5: 0, 1: 1}.
4. Index 2: Sum=29, 29 % 6 = 5. Remainder 5 was seen at index 0. Length = 2 - 0 = 2.
5. Since length >= 2, return true.

Common mistakes candidates make

• Not handling k=0: Modern constraints usually make k >= 1, but if k=0, you cannot use the modulo operator.
• Size constraint: Forgetting the "at least two" rule and returning true for a single element that is a multiple of k.
• Map Initialization: Not including {0: -1} in the map, which prevents the logic from correctly identifying multiples of k that start from the beginning of the array.

Interview preparation tip

Remember: "Same remainder implies difference is a multiple." This is one of the most useful mathematical properties for subarray problems involving divisibility.