LeetCode #2261 — MEDIUM

K Divisible Elements Subarrays

Move from brute-force thinking to an efficient approach using array strategy.

Solve on LeetCode
The Problem

Problem Statement

Given an integer array nums and two integers k and p, return the number of distinct subarrays, which have at most k elements that are divisible by p.

Two arrays nums1 and nums2 are said to be distinct if:

  • They are of different lengths, or
  • There exists at least one index i where nums1[i] != nums2[i].

A subarray is defined as a non-empty contiguous sequence of elements in an array.

Example 1:

Input: nums = [2,3,3,2,2], k = 2, p = 2
Output: 11
Explanation:
The elements at indices 0, 3, and 4 are divisible by p = 2.
The 11 distinct subarrays which have at most k = 2 elements divisible by 2 are:
[2], [2,3], [2,3,3], [2,3,3,2], [3], [3,3], [3,3,2], [3,3,2,2], [3,2], [3,2,2], and [2,2].
Note that the subarrays [2] and [3] occur more than once in nums, but they should each be counted only once.
The subarray [2,3,3,2,2] should not be counted because it has 3 elements that are divisible by 2.

Example 2:

Input: nums = [1,2,3,4], k = 4, p = 1
Output: 10
Explanation:
All element of nums are divisible by p = 1.
Also, every subarray of nums will have at most 4 elements that are divisible by 1.
Since all subarrays are distinct, the total number of subarrays satisfying all the constraints is 10.

Constraints:

  • 1 <= nums.length <= 200
  • 1 <= nums[i], p <= 200
  • 1 <= k <= nums.length

Follow up:

Can you solve this problem in O(n2) time complexity?

Patterns Used

Roadmap

  1. Brute Force Baseline
  2. Core Insight
  3. Algorithm Walkthrough
  4. Edge Cases
  5. Full Annotated Code
  6. Interactive Study Demo
  7. Complexity Analysis
Step 01

Brute Force Baseline

Problem summary: Given an integer array nums and two integers k and p, return the number of distinct subarrays, which have at most k elements that are divisible by p. Two arrays nums1 and nums2 are said to be distinct if: They are of different lengths, or There exists at least one index i where nums1[i] != nums2[i]. A subarray is defined as a non-empty contiguous sequence of elements in an array.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Array · Hash Map · Trie

Example 1

[2,3,3,2,2]
2
2

Example 2

[1,2,3,4]
4
1

Related Problems

  • Subarrays with K Different Integers (subarrays-with-k-different-integers)
  • Count Number of Nice Subarrays (count-number-of-nice-subarrays)
  • Subarray With Elements Greater Than Varying Threshold (subarray-with-elements-greater-than-varying-threshold)
Step 02

Core Insight

What unlocks the optimal approach

  • Enumerate all subarrays and find the ones that satisfy all the conditions.
  • Use any suitable method to hash the subarrays to avoid duplicates.
Interview move: turn each hint into an invariant you can check after every iteration/recursion step.
Step 03

Algorithm Walkthrough

Iteration Checklist

  1. Define state (indices, window, stack, map, DP cell, or recursion frame).
  2. Apply one transition step and update the invariant.
  3. Record answer candidate when condition is met.
  4. Continue until all input is consumed.
Use the first example testcase as your mental trace to verify each transition.
Step 04

Edge Cases

Minimum Input
Single element / shortest valid input
Validate boundary behavior before entering the main loop or recursion.
Duplicates & Repeats
Repeated values / repeated states
Decide whether duplicates should be merged, skipped, or counted explicitly.
Extreme Constraints
Upper-end input sizes
Re-check complexity target against constraints to avoid time-limit issues.
Invalid / Corner Shape
Empty collections, zeros, or disconnected structures
Handle special-case structure before the core algorithm path.
Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #2261: K Divisible Elements Subarrays
class Solution {
    public int countDistinct(int[] nums, int k, int p) {
        Set<Long> s = new HashSet<>();
        int n = nums.length;
        int base1 = 131, base2 = 13331;
        int mod1 = (int) 1e9 + 7, mod2 = (int) 1e9 + 9;
        for (int i = 0; i < n; ++i) {
            long h1 = 0, h2 = 0;
            int cnt = 0;
            for (int j = i; j < n; ++j) {
                cnt += nums[j] % p == 0 ? 1 : 0;
                if (cnt > k) {
                    break;
                }
                h1 = (h1 * base1 + nums[j]) % mod1;
                h2 = (h2 * base2 + nums[j]) % mod2;
                s.add(h1 << 32 | h2);
            }
        }
        return s.size();
    }
}
Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Press Step or Run All to begin.
Step 07

Complexity Analysis

Time
O(n^2)
Space
O(n^2)

Approach Breakdown

HASH SET
O(N × L) time
O(N × L) space

Store all N words in a hash set. Each insert/lookup hashes the entire word of length L, giving O(L) per operation. Prefix queries require checking every stored word against the prefix — O(N × L) per prefix search. Space is O(N × L) for storing all characters.

TRIE
O(L) time
O(N × L) space

Each operation (insert, search, prefix) takes O(L) time where L is the word length — one node visited per character. Total space is bounded by the sum of all stored word lengths. Tries win over hash sets when you need prefix matching: O(L) prefix search vs. checking every stored word.

Shortcut: One node per character → O(L) per operation. Prefix queries are what make tries worthwhile.
Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.

Mutating counts without cleanup

Wrong move: Zero-count keys stay in map and break distinct/count constraints.

Usually fails on: Window/map size checks are consistently off by one.

Fix: Delete keys when count reaches zero.