1. What is Remainder?

The remainder is the amount left over after performing division of two integers when the dividend isn't a multiple of the divisor. It's a fundamental concept in modular arithmetic and number theory.

2. How Does the Calculator Work?

The calculator uses the remainder formula:

Where:

• \( a \) — Dividend (the number being divided)
• \( b \) — Divisor (the number dividing by)
• \( \left\lfloor \frac{a}{b} \right\rfloor \) — Floor of the division (greatest integer less than or equal to a/b)

Explanation: The formula calculates how much is left after subtracting the largest multiple of b that fits into a.

3. Importance of Remainder Calculation

Details: Remainders are essential in computer science (hashing, cryptography), calendar calculations, timekeeping, and solving divisibility problems in mathematics.

4. Using the Calculator

Tips: Enter any real number for dividend (a) and any non-zero real number for divisor (b). The calculator will compute a mod b.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between remainder and modulus?
A: For positive numbers they're identical. For negative numbers, remainder preserves the dividend's sign while modulus is always positive.

Q2: Can the divisor be negative?
A: Yes, but the result will be the same as with a positive divisor since floor division handles negative numbers appropriately.

Q3: What's the remainder when dividing by 1?
A: Always 0, since any integer is exactly divisible by 1 with no remainder.

Q4: How is remainder used in programming?
A: Most languages have a modulus operator (%) that calculates remainder, used for cyclic operations, array indexing, and more.

Q5: What about remainder with floating-point numbers?
A: The same formula applies - the result is the exact remainder after division, which may be a non-integer value.