Sum Multiples - Practice Coding Problems

Sum Multiples - Problem

Math Easy

Sum Multiples is a classic mathematical problem that appears frequently in coding interviews.

You're given a positive integer n, and your task is to find the sum of all integers in the range [1, n] inclusive that are divisible by at least one of these numbers: 3, 5, or 7.

For example, if n = 10, the numbers divisible by 3, 5, or 7 are: 3, 5, 6, 7, 9, 10. Their sum is 3 + 5 + 6 + 7 + 9 + 10 = 40.

Goal: Return an integer representing the sum of all valid numbers.
Input: A positive integer n
Output: The sum of all multiples of 3, 5, or 7 in range [1, n]

Input & Output

example_1.py — Basic Case

$ Input: n = 7

› Output: 21

💡 Note: Numbers divisible by 3, 5, or 7 in range [1,7]: 3, 5, 6, 7. Sum = 3 + 5 + 6 + 7 = 21

example_2.py — Larger Range

$ Input: n = 10

› Output: 40

💡 Note: Numbers divisible by 3, 5, or 7 in range [1,10]: 3, 5, 6, 7, 9, 10. Sum = 3 + 5 + 6 + 7 + 9 + 10 = 40

example_3.py — Edge Case

$ Input: n = 3

› Output: 3

💡 Note: Only number 3 is divisible by 3, 5, or 7 in range [1,3]. Sum = 3

Constraints

1 ≤ n ≤ 10⁹
n is a positive integer

Visualization

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Understanding the Visualization

Identify the Pattern

Recognize that we need multiples of 3, 5, or 7

Apply Set Theory

Use inclusion-exclusion to handle overlapping multiples

Use Arithmetic Series

Calculate sums with formula: n×(n+1)÷2 for sequence 1,2,3...n

Achieve O(1)

Replace iteration with direct mathematical calculation

Key Takeaway

🎯 Key Insight: Mathematical formulas can replace iterative algorithms, transforming O(n) problems into O(1) solutions using principles like inclusion-exclusion and arithmetic series.

Asked in

G Google 15 a Amazon 12 f Meta 8 ⊞ Microsoft 10

The optimal solution uses the inclusion-exclusion principle with arithmetic series formulas to achieve O(1) time complexity. Instead of iterating through numbers, we calculate sums mathematically: add multiples of 3, 5, and 7, subtract overlapping multiples of 15, 21, and 35, then add back the triple overlap of 105.

Common Approaches

Approach	Time	Space	Notes
✓ Mathematical Formula (Optimal)	O(1)	O(1)	Use inclusion-exclusion principle with arithmetic series formulas
Brute Force (Linear Iteration)	O(n)	O(1)	Check every number from 1 to n for divisibility by 3, 5, or 7

Mathematical Formula (Optimal) — Algorithm Steps

Calculate sum of multiples of 3 using arithmetic series formula
Calculate sum of multiples of 5 using arithmetic series formula
Calculate sum of multiples of 7 using arithmetic series formula
Subtract overlaps: sum of multiples of 15, 21, and 35
Add back triple overlap: sum of multiples of 105
Return the final result

Visualization

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Step-by-Step Walkthrough

Individual Sets

Calculate sum of multiples for each divisor (3, 5, 7)

Remove Overlaps

Subtract double-counted elements (15, 21, 35)

Add Triple Overlap

Add back the triple-subtracted elements (105)

Code -

solution.c — C

#include <stdio.h>

long long sumOfMultiplesOf(int divisor, int limit) {
    long long count = limit / divisor;
    return divisor * count * (count + 1) / 2;
}

long long sumOfMultiples(int n) {
    return sumOfMultiplesOf(3, n) +
           sumOfMultiplesOf(5, n) +
           sumOfMultiplesOf(7, n) -
           sumOfMultiplesOf(15, n) -
           sumOfMultiplesOf(21, n) -
           sumOfMultiplesOf(35, n) +
           sumOfMultiplesOf(105, n);
}

int main() {
    int n;
    scanf("%d", &n);
    printf("%lld\n", sumOfMultiples(n));
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(1)

All operations are mathematical calculations that take constant time regardless of input size

✓ Linear Growth

Space Complexity

O(1)

Only using a constant amount of extra space for calculations

✓ Linear Space

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Mathematical Formula (Optimal) — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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