LeetCode #2961 — MEDIUM

Double Modular Exponentiation

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

The Problem

Problem Statement

You are given a 0-indexed 2D array variables where variables[i] = [a_i, b_i, c_i, m_i], and an integer target.

An index i is good if the following formula holds:

0 <= i < variables.length
((a_i^b_i % 10)^c_i) % m_i == target

Return an array consisting of good indices in any order.

Example 1:

Input: variables = [[2,3,3,10],[3,3,3,1],[6,1,1,4]], target = 2
Output: [0,2]
Explanation: For each index i in the variables array:
1) For the index 0, variables[0] = [2,3,3,10], (2³ % 10)³ % 10 = 2.
2) For the index 1, variables[1] = [3,3,3,1], (3³ % 10)³ % 1 = 0.
3) For the index 2, variables[2] = [6,1,1,4], (6¹ % 10)¹ % 4 = 2.
Therefore we return [0,2] as the answer.

Example 2:

Input: variables = [[39,3,1000,1000]], target = 17
Output: []
Explanation: For each index i in the variables array:
1) For the index 0, variables[0] = [39,3,1000,1000], (39³ % 10)¹⁰⁰⁰ % 1000 = 1.
Therefore we return [] as the answer.

Constraints:

1 <= variables.length <= 100
variables[i] == [a_i, b_i, c_i, m_i]
1 <= a_i, b_i, c_i, m_i <= 10³
0 <= target <= 10³

Step 01

Brute Force Baseline

Problem summary: You are given a 0-indexed 2D array variables where variables[i] = [ai, bi, ci, mi], and an integer target. An index i is good if the following formula holds: 0 <= i < variables.length ((aibi % 10)ci) % mi == target Return an array consisting of good indices in any order.

Baseline thinking

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

Pattern signal: Array · Math

Example 1

[[2,3,3,10],[3,3,3,1],[6,1,1,4]]
2

Example 2

[[39,3,1000,1000]]
17

Step 02

Core Insight

What unlocks the optimal approach

No official hints in dataset. Start from constraints and look for a monotonic or reusable state.

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

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 #2961: Double Modular Exponentiation
class Solution {
    public List<Integer> getGoodIndices(int[][] variables, int target) {
        List<Integer> ans = new ArrayList<>();
        for (int i = 0; i < variables.length; ++i) {
            var e = variables[i];
            int a = e[0], b = e[1], c = e[2], m = e[3];
            if (qpow(qpow(a, b, 10), c, m) == target) {
                ans.add(i);
            }
        }
        return ans;
    }

    private int qpow(long a, int n, int mod) {
        long ans = 1;
        for (; n > 0; n >>= 1) {
            if ((n & 1) == 1) {
                ans = ans * a % mod;
            }
            a = a * a % mod;
        }
        return (int) ans;
    }
}

// Accepted solution for LeetCode #2961: Double Modular Exponentiation
func getGoodIndices(variables [][]int, target int) (ans []int) {
	qpow := func(a, n, mod int) int {
		ans := 1
		for ; n > 0; n >>= 1 {
			if n&1 == 1 {
				ans = ans * a % mod
			}
			a = a * a % mod
		}
		return ans
	}
	for i, e := range variables {
		a, b, c, m := e[0], e[1], e[2], e[3]
		if qpow(qpow(a, b, 10), c, m) == target {
			ans = append(ans, i)
		}
	}
	return
}

# Accepted solution for LeetCode #2961: Double Modular Exponentiation
class Solution:
    def getGoodIndices(self, variables: List[List[int]], target: int) -> List[int]:
        return [
            i
            for i, (a, b, c, m) in enumerate(variables)
            if pow(pow(a, b, 10), c, m) == target
        ]

// Accepted solution for LeetCode #2961: Double Modular Exponentiation
// 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 #2961: Double Modular Exponentiation
// class Solution {
//     public List<Integer> getGoodIndices(int[][] variables, int target) {
//         List<Integer> ans = new ArrayList<>();
//         for (int i = 0; i < variables.length; ++i) {
//             var e = variables[i];
//             int a = e[0], b = e[1], c = e[2], m = e[3];
//             if (qpow(qpow(a, b, 10), c, m) == target) {
//                 ans.add(i);
//             }
//         }
//         return ans;
//     }
// 
//     private int qpow(long a, int n, int mod) {
//         long ans = 1;
//         for (; n > 0; n >>= 1) {
//             if ((n & 1) == 1) {
//                 ans = ans * a % mod;
//             }
//             a = a * a % mod;
//         }
//         return (int) ans;
//     }
// }

// Accepted solution for LeetCode #2961: Double Modular Exponentiation
function getGoodIndices(variables: number[][], target: number): number[] {
    const qpow = (a: number, n: number, mod: number) => {
        let ans = 1;
        for (; n; n >>= 1) {
            if (n & 1) {
                ans = Number((BigInt(ans) * BigInt(a)) % BigInt(mod));
            }
            a = Number((BigInt(a) * BigInt(a)) % BigInt(mod));
        }
        return ans;
    };
    const ans: number[] = [];
    for (let i = 0; i < variables.length; ++i) {
        const [a, b, c, m] = variables[i];
        if (qpow(qpow(a, b, 10), c, m) === target) {
            ans.push(i);
        }
    }
    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.

Overflow in intermediate arithmetic

Wrong move: Temporary multiplications exceed integer bounds.

Usually fails on: Large inputs wrap around unexpectedly.

Fix: Use wider types, modular arithmetic, or rearranged operations.