1. What is Modulus Operation?

The modulus operation finds the remainder after division of one number by another. For example, 2 mod 5 = 2 because when 2 is divided by 5, the remainder is 2.

2. How Does Modulus Work?

The modulus operation follows this mathematical formula:

Where:

• \( a \) — The dividend (number being divided)
• \( b \) — The divisor (number dividing by)
• The result is always between 0 and \( b-1 \)

Examples:

• 7 mod 3 = 1 (7 ÷ 3 = 2 with remainder 1)
• 10 mod 4 = 2 (10 ÷ 4 = 2 with remainder 2)
• 2 mod 5 = 2 (2 ÷ 5 = 0 with remainder 2)

3. Importance of Modulus

Applications: Modulus operations are fundamental in computer programming, cryptography, time calculations, and determining even/odd numbers.

4. Using the Calculator

Instructions: Enter two numbers (a and b) where b ≠ 0. The calculator will compute a mod b (the remainder when a is divided by b).

5. Frequently Asked Questions (FAQ)

Q1: What happens if b is 0?
A: Division by zero is undefined, so b must be a non-zero number.

Q2: Can modulus be negative?
A: In mathematics, the remainder is always non-negative. Some programming languages may handle negative numbers differently.

Q3: What's the difference between modulus and remainder?
A: Mathematically they're the same, but some programming languages implement them differently with negative numbers.

Q4: What is 2 mod 5?
A: 2 mod 5 = 2, as shown in the calculator's default example.

Q5: Where is modulus used in real life?
A: Common uses include clock arithmetic (12-hour time), calendar calculations, and hash functions in computer science.