1. What is Modulo Operation?

The modulo operation finds the remainder after division of one number by another. Given two positive numbers, a (the dividend) and b (the divisor), a modulo b is the remainder of the Euclidean division of a by b.

2. How Does the Calculator Work?

The calculator uses the modulo formula:

Where:

• \( a \) — Dividend (the number being divided)
• \( b \) — Divisor (the number dividing by)
• \( \left\lfloor \frac{a}{b} \right\rfloor \) — Floor function of the division

Explanation: The modulo operation returns the remainder after division. For example, 7 mod 3 would be 1, since 3 goes into 7 twice (3 × 2 = 6) with a remainder of 1.

3. Importance of Modulo Calculation

Details: Modulo operations are fundamental in computer science, cryptography, and mathematics. They're used for hashing, generating random numbers, circular arrays, and determining even/odd numbers.

4. Using the Calculator

Tips: Enter the dividend (a) and divisor (b) values. The divisor must be non-zero. 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 identical. For negative numbers, modulo always returns a positive result between 0 and b-1, while remainder can be negative.

Q2: Can the divisor be zero?
A: No, division by zero is undefined in mathematics, so modulo by zero is also undefined.

Q3: What are practical uses of modulo?
A: Common uses include determining if a number is even/odd (n mod 2), wrapping values in circular buffers, and in cryptographic algorithms.

Q4: How does modulo work with negative numbers?
A: This calculator implements "floored division" modulo where the result has the same sign as the divisor. For example, -5 mod 3 is 1.

Q5: Is modulo the same in all programming languages?
A: No, some languages use truncated division (like C) which gives different results with negative numbers than floored division used here.