Sum Pairwise Remainders Across Two Integer Ranges
Company: IBM
Role: Software Engineer
Category: Coding & Algorithms
Difficulty: easy
Interview Round: Online Assessment
# Sum Pairwise Remainders Across Two Integer Ranges
You are given two non-negative integers `upperX` and `upperY`. Return the sum of `x mod y` over every integer `x` in `[0, upperX]` and every valid divisor `y` in `[1, upperY]`.
Implement `sumRangeRemainders(upperX, upperY)`.
## Input and Output
- Return $\sum_{x=0}^{\text{upperX}}\sum_{y=1}^{\text{upperY}}(x\bmod y)$.
- The original range description includes zero at both lower bounds. Because modulo by zero is undefined, this practice version excludes `y = 0` while retaining `x = 0`.
## Constraints
- `0 <= upperX <= 1,000,000`
- `1 <= upperY <= 1,000,000`
- The result fits in a signed 64-bit integer.
- Iterating over all `(upperX + 1) * upperY` pairs will not finish within the intended limits.
## Example 1
```text
Input: upperX = 2, upperY = 2
Output: 1
```
For divisors `1` and `2`, the remainder totals are `0` and `1`.
## Example 2
```text
Input: upperX = 3, upperY = 3
Output: 5
```
The totals for divisors `1`, `2`, and `3` are `0`, `2`, and `3`.
Quick Answer: Sum every remainder produced by pairing an integer in one inclusive range with each valid divisor in another. This algorithmic exercise emphasizes quotient-block aggregation, the undefined zero-divisor boundary, and 64-bit arithmetic without a quadratic loop.