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 m (the divisor), a modulo m is the remainder of the Euclidean division of a by m.

2. How Does the Calculator Work?

The calculator uses the modulo formula:

Where:

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

Explanation: The formula calculates how many times m fits completely into a (the quotient), then returns what's left over (the remainder).

3. Importance of Modulo Calculation

Details: Modulo operations are fundamental in computer science (hashing, cryptography), mathematics (number theory), and engineering (signal processing). They're used for cyclic operations and pattern detection.

4. Using the Calculator

Tips: Enter any real number for the dividend (a) and any non-zero real number for the divisor (m). The calculator handles both positive and negative numbers correctly.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between modulo and remainder?
A: For positive numbers they're the same, but for negative numbers, modulo always returns a positive result between 0 and m-1, while remainder can be negative.

Q2: Can the divisor (m) be negative?
A: Yes, but the result will always have the same sign as the divisor. However, most applications use positive divisors.

Q3: What happens if m is zero?
A: Division by zero is undefined, so modulo with zero divisor is also undefined. The calculator prevents this.

Q4: How is modulo used in programming?
A: Common uses include checking even/odd (n mod 2), array indexing, cryptography algorithms, and generating random numbers.

Q5: What's the behavior for fractional numbers?
A: The calculator works with fractional numbers, returning the fractional remainder. For example, 5.5 mod 2.1 = 1.3.