LeetCode #3729 — HARD

Count Distinct Subarrays Divisible by K in Sorted Array

Break down a hard problem into reliable checkpoints, edge-case handling, and complexity trade-offs.

The Problem

Problem Statement

You are given an integer array nums sorted in non-descending order and a positive integer k.

A subarray of nums is good if the sum of its elements is divisible by k.

Return an integer denoting the number of distinct good subarrays of nums.

Subarrays are distinct if their sequences of values are. For example, there are 3 distinct subarrays in [1, 1, 1], namely [1], [1, 1], and [1, 1, 1].

Example 1:

Input: nums = [1,2,3], k = 3

Output: 3

Explanation:

The good subarrays are [1, 2], [3], and [1, 2, 3]. For example, [1, 2, 3] is good because the sum of its elements is 1 + 2 + 3 = 6, and 6 % k = 6 % 3 = 0.

Example 2:

Input: nums = [2,2,2,2,2,2], k = 6

Output: 2

Explanation:

The good subarrays are [2, 2, 2] and [2, 2, 2, 2, 2, 2]. For example, [2, 2, 2] is good because the sum of its elements is 2 + 2 + 2 = 6, and 6 % k = 6 % 6 = 0.

Note that [2, 2, 2] is counted only once.

Constraints:

1 <= nums.length <= 10⁵
1 <= nums[i] <= 10⁹
nums is sorted in non-descending order.
1 <= k <= 10⁹

Step 01

Brute Force Baseline

Problem summary: You are given an integer array nums sorted in non-descending order and a positive integer k. A subarray of nums is good if the sum of its elements is divisible by k. Return an integer denoting the number of distinct good subarrays of nums. Subarrays are distinct if their sequences of values are. For example, there are 3 distinct subarrays in [1, 1, 1], namely [1], [1, 1], and [1, 1, 1].

Baseline thinking

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

Pattern signal: Array · Hash Map

Example 1

[1,2,3]
3

Example 2

[2,2,2,2,2,2]
6

Step 02

Core Insight

What unlocks the optimal approach

Use prefix sums.
Any subarray with >= 2 distinct values has a unique strict-increase boundary <code>(i, j)</code>, so the value sequence is identified by that boundary and how many elements you take from each edge.
Count single-value sequences per run by testing <code>(L*val) % k == 0</code>.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
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

Largest constraint values

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 #3729: Count Distinct Subarrays Divisible by K in Sorted Array
class Solution {
    public long numGoodSubarrays(int[] nums, int k) {
        long ans = 0;
        int s = 0;
        Map<Integer, Integer> cnt = new HashMap<>();
        cnt.put(0, 1);
        for (int x : nums) {
            s = (s + x) % k;
            ans += cnt.getOrDefault(s, 0);
            cnt.merge(s, 1, Integer::sum);
        }
        int n = nums.length;
        for (int i = 0; i < n;) {
            int j = i + 1;
            while (j < n && nums[j] == nums[i]) {
                ++j;
            }
            int m = j - i;
            for (int h = 1; h <= m; ++h) {
                if (1L * nums[i] * h % k == 0) {
                    ans -= (m - h);
                }
            }
            i = j;
        }
        return ans;
    }
}

// Accepted solution for LeetCode #3729: Count Distinct Subarrays Divisible by K in Sorted Array
func numGoodSubarrays(nums []int, k int) (ans int64) {
    s := 0
    cnt := map[int]int{0: 1}
    for _, x := range nums {
        s = (s + x) % k
        ans += int64(cnt[s])
        cnt[s]++
    }

    n := len(nums)
    for i := 0; i < n; {
        j := i + 1
        for j < n && nums[j] == nums[i] {
            j++
        }
        m := j - i
        for h := 1; h <= m; h++ {
            if int64(nums[i])*int64(h)%int64(k) == 0 {
                ans -= int64(m - h)
            }
        }
        i = j
    }
    return
}

# Accepted solution for LeetCode #3729: Count Distinct Subarrays Divisible by K in Sorted Array
class Solution:
    def numGoodSubarrays(self, nums: List[int], k: int) -> int:
        cnt = Counter({0: 1})
        ans = s = 0
        for x in nums:
            s = (s + x) % k
            ans += cnt[s]
            cnt[s] += 1
        n = len(nums)
        i = 0
        while i < n:
            j = i + 1
            while j < n and nums[j] == nums[i]:
                j += 1
            m = j - i
            for h in range(1, m + 1):
                if (h * nums[i]) % k == 0:
                    ans -= m - h
            i = j
        return ans

// Accepted solution for LeetCode #3729: Count Distinct Subarrays Divisible by K in Sorted Array
// Rust example auto-generated from java reference.
// Replace the signature and local types with the exact LeetCode harness for this problem.
impl Solution {
    pub fn rust_example() {
        // Port the logic from the reference block below.
    }
}

// Reference (java):
// // Accepted solution for LeetCode #3729: Count Distinct Subarrays Divisible by K in Sorted Array
// class Solution {
//     public long numGoodSubarrays(int[] nums, int k) {
//         long ans = 0;
//         int s = 0;
//         Map<Integer, Integer> cnt = new HashMap<>();
//         cnt.put(0, 1);
//         for (int x : nums) {
//             s = (s + x) % k;
//             ans += cnt.getOrDefault(s, 0);
//             cnt.merge(s, 1, Integer::sum);
//         }
//         int n = nums.length;
//         for (int i = 0; i < n;) {
//             int j = i + 1;
//             while (j < n && nums[j] == nums[i]) {
//                 ++j;
//             }
//             int m = j - i;
//             for (int h = 1; h <= m; ++h) {
//                 if (1L * nums[i] * h % k == 0) {
//                     ans -= (m - h);
//                 }
//             }
//             i = j;
//         }
//         return ans;
//     }
// }

// Accepted solution for LeetCode #3729: Count Distinct Subarrays Divisible by K in Sorted Array
function numGoodSubarrays(nums: number[], k: number): number {
    let ans = 0;
    let s = 0;
    const cnt = new Map<number, number>();
    cnt.set(0, 1);

    for (const x of nums) {
        s = (s + x) % k;
        ans += cnt.get(s) ?? 0;
        cnt.set(s, (cnt.get(s) ?? 0) + 1);
    }

    const n = nums.length;
    for (let i = 0; i < n; ) {
        let j = i + 1;
        while (j < n && nums[j] === nums[i]) ++j;
        const m = j - i;
        for (let h = 1; h <= m; ++h) {
            if ((nums[i] * h) % k === 0) {
                ans -= m - h;
            }
        }
        i = j;
    }

    return ans;
}

Step 06

Interactive Study Demo

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

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(n)

Space

O(1)

Approach Breakdown

BRUTE FORCE

O(n²) time

O(1) space

Two nested loops check every pair or subarray. The outer loop fixes a starting point, the inner loop extends or searches. For n elements this gives up to n²/2 operations. No extra space, but the quadratic time is prohibitive for large inputs.

OPTIMIZED

O(n) time

O(1) space

Most array problems have an O(n²) brute force (nested loops) and an O(n) optimal (single pass with clever state tracking). The key is identifying what information to maintain as you scan: a running max, a prefix sum, a hash map of seen values, or two pointers.

Shortcut: If you are using nested loops on an array, there is almost always an O(n) solution. Look for the right auxiliary state.

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.