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| 1 | +# XOR After Range Multiplication Queries I (Medium) |
| 2 | + |
| 3 | +**Problem ID:** 3653 |
| 4 | +**Date:** 2026-04-08 |
| 5 | +**Link:** https://leetcode.com/problems/xor-after-range-multiplication-queries-i/ |
| 6 | + |
| 7 | +## Approach |
| 8 | + |
| 9 | +To solve the problem "XOR After Range Multiplication Queries I," we can follow a systematic approach that efficiently processes the queries and computes the final XOR of the modified array. |
| 10 | + |
| 11 | +### Main Idea: |
| 12 | +The goal is to apply a series of multiplication operations specified in the queries to the elements of the array `nums`, and then compute the XOR of the entire array after all operations have been applied. Given the constraints, a naive approach that directly modifies the array for each query could lead to inefficiencies, particularly when the number of queries and the size of the array are both large. |
| 13 | + |
| 14 | +### Approach: |
| 15 | +1. **Iterate Through Queries**: For each query, which consists of four parameters `[l_i, r_i, k_i, v_i]`, we will: |
| 16 | + - Start from index `l_i` and apply the multiplication operation to elements at indices `l_i`, `l_i + k_i`, `l_i + 2*k_i`, ..., up to `r_i`. |
| 17 | + - This means we will update `nums[idx]` to `(nums[idx] * v_i) % (10^9 + 7)` for each valid index in the specified range. |
| 18 | + |
| 19 | +2. **Efficient Indexing**: Instead of updating each element in a nested loop, we can directly compute the indices to be updated using the formula: |
| 20 | + - `idx = l_i + j * k_i` where `j` is a non-negative integer such that `idx` remains within the bounds of `r_i`. |
| 21 | + |
| 22 | +3. **Final XOR Calculation**: After processing all the queries, we will compute the XOR of all elements in the modified `nums` array. This can be done in a single pass through the array. |
| 23 | + |
| 24 | +### Data Structures: |
| 25 | +- We primarily use an integer array `nums` to store the values and a list (or array) to hold the queries. |
| 26 | + |
| 27 | +### Complexity: |
| 28 | +- **Time Complexity**: The overall time complexity is O(q * (r_i - l_i) / k_i), where `q` is the number of queries. In the worst case, this can approach O(n * q) if every query affects most of the elements in the array. However, since the maximum constraints are relatively small (both `n` and `q` up to 1000), this is manageable. |
| 29 | +- **Space Complexity**: O(1) additional space is used aside from the input arrays, as we are modifying `nums` in place and using a few variables for computations. |
| 30 | + |
| 31 | +By following this approach, we efficiently apply the range multiplication operations and compute the final XOR without unnecessary overhead, ensuring that we stay within acceptable time limits for the given constraints. |
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