Remainder Calculator

Find the remainder of a division problem

Enter an integer dividend and divisor to find the remainder left over after the division. This is also known as the modulo operation.

Other Titles
Understanding Remainders: A Comprehensive Guide
Explore the components of division—dividend, divisor, quotient, and remainder—and understand how they relate to each other.

What is a Remainder?

In arithmetic, the remainder is the integer amount 'left over' after dividing one integer by another to produce an integer quotient. It's what remains when the divisor doesn't fit perfectly into the dividend.
The division operation involves four key terms:
These terms are related by the Division Algorithm equation:
Dividend = (Divisor × Quotient) + Remainder

Calculation Example

  • Let's divide 17 by 5.
  • **Dividend:** 17
  • **Divisor:** 5
  • How many times does 5 go into 17? It goes in 3 times (5 × 3 = 15). So, the quotient is 3.
  • What is left over? 17 - 15 = 2.
  • **Remainder:** 2
  • Checking the formula: 17 = (5 × 3) + 2 => 17 = 15 + 2. This is correct.

Step-by-Step Guide to Using the Remainder Calculator

This calculator quickly provides the remainder for any integer division.
How to Use It:

Usage Tips

  • This calculator is designed for integer division.
  • The divisor cannot be zero, as division by zero is undefined.
  • The operation performed is often called the 'modulo' or 'modulus' operation.

Real-World Applications of Remainders (Modulo Arithmetic)

The remainder operation, or modulo arithmetic, is incredibly useful in programming, computer science, and everyday life.
Programming and Computer Science:
Everyday Life:

Practical Examples

  • What time will it be 50 hours from now? Since a day has 24 hours, we calculate `50 % 24`. The remainder is 2. So, it will be 2 hours past the current time, two days from now.
  • A programmer wants to color every 3rd row in a spreadsheet. They can check `if (row_number % 3 === 0)`. The rows with a remainder of 0 will be colored.

Common Misconceptions and Correct Methods

Misconception: The Remainder is the Decimal Part
When a calculator shows 17 / 5 = 3.4, the remainder is not 4. The remainder is an integer. To find it from the decimal, you multiply the decimal part by the divisor: 0.4 * 5 = 2.
Dealing with Negative Numbers
How remainders for negative numbers are calculated can vary between programming languages. However, the mathematical definition states that the remainder r must be non-negative and satisfy 0 <= r < |divisor|. For example, -17 mod 5 could be 3, because -17 = 5 * (-4) + 3. This calculator uses the % operator common in many programming languages, where the sign of the remainder typically matches the sign of the dividend.

Key Takeaways

  • The remainder is the 'what's left over' part of an integer division.
  • It's also known as the modulo operation.
  • The remainder is always smaller than the divisor.

Mathematical Derivation and Examples

The relationship between the four parts of a division is formally known as the Division Algorithm.
The Division Algorithm
For any two integers, a (the dividend) and b (the divisor), with b ≠ 0, there exist unique integers q (the quotient) and r (the remainder) such that:
a = b * q + r
and 0 <= r < |b|.
This theorem is the formal basis for all integer division and guarantees that for any pair of integers, a unique quotient and remainder exist.

Comprehensive Example

  • Let's find the remainder of -52 divided by 10.
  • **a = -52, b = 10**
  • We want to find `r`.
  • Many programming languages (and this calculator) would compute `-52 % 10` as -2. This is because `q` is taken by truncating towards zero (-5), so `-52 = 10 * (-5) + (-2)`.
  • In a strictly mathematical context (where remainder must be positive), the quotient `q` would be -6. Then, `-52 = 10 * (-6) + 8`. In this case, the remainder `r` would be 8.