1. What is Division With Remainder?

Division with remainder, also known as Euclidean division, is the process of dividing two integers which produces a quotient and a remainder. This is fundamental in arithmetic and number theory.

2. How Does the Calculator Work?

The calculator uses the division algorithm:

Where:

• \( a \) — Dividend (number to be divided)
• \( b \) — Divisor (number to divide by)
• \( q \) — Quotient (integer part of division)
• \( r \) — Remainder (must satisfy 0 ≤ r < b)

Explanation: The algorithm guarantees that for any integers a and b (b ≠ 0), there exist unique integers q and r satisfying the equation with 0 ≤ r < |b|.

3. Applications of Division With Remainder

Details: This concept is used in computer programming (modulo operation), cryptography, calendar calculations, and determining whether numbers are even or odd.

4. Using the Calculator

Tips: Enter the dividend and divisor (must be positive integers). The calculator will show the quotient, remainder, and the complete equation.

5. Frequently Asked Questions (FAQ)

Q1: What happens if the divisor is zero?
A: Division by zero is undefined. The calculator requires the divisor to be at least 1.

Q2: Can the remainder be zero?
A: Yes, when the dividend is exactly divisible by the divisor (e.g., 10 ÷ 2 = 5 R0).

Q3: How is this different from floating-point division?
A: This gives integer results with remainder, while floating-point division gives fractional results.

Q4: What's the maximum remainder possible?
A: The remainder is always less than the divisor (r < b).

Q5: How is this used in programming?
A: The modulo operation (a % b) is widely used for hashing, circular buffers, and determining divisibility.