Remainder Calculator

What is Integer Division Remainder?

When you divide one integer by another, the remainder is what's left over after the division is complete. Using the division algorithm, for any integers a (dividend) and d (divisor, d > 0), there exist unique integers q (quotient) and r (remainder) such that a = d × q + r, where 0 ≤ r < d.

How to Use This Calculator

1. Enter the Dividend: The number you want to divide
2. Enter the Divisor: The number to divide by (cannot be zero)
3. View Results: See the quotient, remainder, and step-by-step solution
4. Try Examples: Use quick examples to understand the concept

The Division Algorithm

Formula: a = d × q + r

• a = dividend (the number being divided)
• d = divisor (the number to divide by)
• q = quotient (the integer result of division)
• r = remainder (what's left over, where 0 ≤ r < |d|)

Understanding the Relationship

The remainder operation is closely related to the modulo operation. For positive numbers, they produce the same result. For example:

• 17 ÷ 5 = 3 remainder 2 (because 17 = 5 × 3 + 2)
• 23 ÷ 7 = 3 remainder 2 (because 23 = 7 × 3 + 2)
• 100 ÷ 11 = 9 remainder 1 (because 100 = 11 × 9 + 1)

Common Applications

Divisibility Testing

A number is divisible by another if the remainder is 0. For example, 15 ÷ 3 = 5 remainder 0, so 15 is divisible by 3.

Distribution Problems

When distributing items equally, the remainder tells you how many items are left over. For example, distributing 23 candies among 5 children: each gets 4 candies, with 3 remaining.

Time Conversions

Converting minutes to hours and minutes uses remainder. 137 minutes = 2 hours and 17 minutes (137 = 60 × 2 + 17).

Programming

The remainder operation (often using % operator) is fundamental in programming for tasks like determining even/odd numbers, circular array indexing, and hash functions.

Frequently Asked Questions

What is the difference between remainder and modulo?

For positive numbers, remainder and modulo produce the same result. They can differ for negative numbers depending on the definition used (truncated vs. floored division).

What happens when the dividend is smaller than the divisor?

If the dividend is smaller than the divisor, the quotient is 0 and the remainder equals the dividend. For example, 3 ÷ 7 = 0 remainder 3.

Can the divisor be zero?

No, division by zero is undefined. The calculator will show an error if you try to divide by zero.

How do I verify my result?

Multiply the quotient by the divisor and add the remainder. The result should equal the dividend: d × q + r = a.

Note: This calculator uses integer division to find the quotient and remainder. For decimal/fractional results, use a standard division calculator.