1. What is Congruence Modulo?

Two integers a and b are congruent modulo m if they have the same remainder when divided by m, or equivalently, if m divides the difference (a - b). This is written as a ≡ b mod m.

2. How Does the Calculator Work?

The calculator uses the definition of congruence modulo:

Where:

• \( a \) — First integer value
• \( b \) — Second integer value
• \( m \) — Modulus (positive integer)

Explanation: The calculator checks if (a - b) is divisible by m without remainder. If yes, a and b are congruent modulo m.

3. Importance of Congruence Modulo

Details: Congruence modulo is fundamental in number theory, cryptography, computer science, and many areas of mathematics. It's used in hash functions, RSA encryption, and solving Diophantine equations.

4. Using the Calculator

Tips: Enter integer values for a and b, and a positive integer for m. The calculator will determine if a ≡ b mod m.

5. Frequently Asked Questions (FAQ)

Q1: What does a ≡ b mod m mean?
A: It means a and b leave the same remainder when divided by m, or equivalently, m divides (a - b).

Q2: Can m be negative?
A: Mathematically yes, but this calculator uses positive m since a ≡ b mod m is the same as a ≡ b mod (-m).

Q3: What's the difference between mod and ≡ mod?
A: The mod operation gives the remainder, while ≡ mod is a relation showing equivalence under modulo.

Q4: What are some properties of congruence?
A: Congruence is reflexive, symmetric, and transitive. Also, congruences can be added, subtracted, and multiplied.

Q5: How is this used in cryptography?
A: Many cryptographic algorithms like RSA rely on properties of modular arithmetic and congruences.