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 Modulo Work?

The modulo operation can be represented as:

Where:

• \( a \) — Dividend (the number being divided)
• \( m \) — Divisor (the number dividing by)
• The result is always between 0 and m-1

Example: 17 mod 5 = 2, because 17 ÷ 5 = 3 with remainder 2.

3. Applications of Modulo

Details: Modulo operations are fundamental in computer science (hashing, cryptography), mathematics (number theory), and everyday applications (clock arithmetic, calendar calculations).

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What happens with negative numbers?
A: The result is always positive between 0 and m-1. For -17 mod 5, the result is 3.

Q2: What if the divisor is negative?
A: The sign of the divisor doesn't affect the result. 17 mod -5 gives the same result as 17 mod 5.

Q3: How is modulo different from remainder?
A: For positive numbers they're the same, but differ with negative numbers. Modulo always returns a positive result.

Q4: What's the result when divisor is 1?
A: Any number mod 1 is 0, since every integer is divisible by 1 with no remainder.

Q5: Can modulo be used with decimals?
A: Yes, the calculator works with decimal numbers. 5.5 mod 2.1 ≈ 1.3.