Divisible and Non-divisible Sums Difference

Divisible and Non-divisible Sums Difference

What is this problem about?

The Divisible and Non-divisible Sums Difference interview question asks you to calculate the difference between two sums within a range from 1 to n. Specifically, you need to find the sum of all integers in the range [1, n] that are NOT divisible by m, and subtract from it the sum of all integers in the same range that ARE divisible by m. It's a straightforward mathematical problem that requires iterating through a sequence and applying a modulo check.

Why is this asked in interviews?

Companies like Meta and Google use this as an entry-level coding task or a warm-up. It tests basic loop constructs, conditional logic, and simple arithmetic. The Divisible and Non-divisible Sums Difference coding problem also provides an opportunity for candidates to demonstrate knowledge of mathematical formulas, such as the sum of an arithmetic progression, which can optimize the solution from linear time to constant time.

Algorithmic pattern used

This problem can be solved using a simple math interview pattern or a linear scan.

1. Linear Scan: Iterate from 1 to n and maintain two counters: one for numbers divisible by m and one for those that are not.
2. Formulaic Approach (O(1)): Use the formula for the sum of the first n integers: S = n * (n + 1) / 2. Then calculate the sum of numbers divisible by m (which also form an arithmetic progression) and derive the non-divisible sum by subtraction.

Example explanation

Suppose n = 10 and m = 3.

1. Range: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10].
2. Divisible by 3: [3, 6, 9]. Sum (num2) = 3 + 6 + 9 = 18.
3. Not divisible by 3: [1, 2, 4, 5, 7, 8, 10]. Sum (num1) = 1 + 2 + 4 + 5 + 7 + 8 + 10 = 37.
4. Difference: num1 - num2 = 37 - 18 = 19.

Common mistakes candidates make

• Wrong Difference Order: Subtracting the non-divisible sum from the divisible sum instead of the other way around.
• Off-by-one Errors: Not including the boundary value n in the summation.
• Inefficient Loops: Using nested loops or complex logic for a problem that only requires a single pass or a formula.

Interview preparation tip

Whenever you encounter a problem involving sums of ranges with divisibility rules, always think about the arithmetic progression sum formula: Sum = (n/2) * (first_term + last_term). This shows the interviewer that you think about performance even for simple tasks.