1. What is Modulo Operation?

The modulo operation finds the remainder after division of one number by another. It's commonly used in programming, cryptography, and mathematics.

2. How Does Modulo Work?

The modulo operation is calculated using:

Where:

• \( a \) — The dividend (number being divided)
• \( b \) — The divisor (number to divide by)
• \( \left\lfloor \frac{a}{b} \right\rfloor \) — Floor division (greatest integer less than or equal to a/b)

Example: mod(7, 3) = 1 because 7 - 3*floor(7/3) = 7 - 3*2 = 1

3. Practical Applications

Details: Modulo is used in:

• Determining even/odd numbers (mod(n, 2))
• Circular array indexing
• Cryptographic algorithms
• Hash functions
• Time calculations (hours mod 12)

4. Using the Calculator

Tips: Enter any real number for 'a' and any non-zero real number for 'b'. The calculator will compute a mod b.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between modulo and remainder?
A: For positive numbers they're the same, but they differ with negative numbers. Modulo always returns a positive result.

Q2: What happens if b is zero?
A: Division by zero is undefined, so the modulus is also undefined.

Q3: Can modulo be used with decimal numbers?
A: Yes, the modulo operation works with any real numbers (except b=0).

Q4: How is modulo used in programming?
A: Most programming languages have a modulo operator (%), used for cyclic operations and checking divisibility.

Q5: What's the range of results from modulo?
A: The result is always in the range [0, b) for positive b.