Quadratic Equation Solver (ax² + bx + c = 0)
About Modulo Calculator
The Modular Arithmetic & Congruence Calculator computes remainder operations ($A \bmod B$), Modular Exponentiation ($A^B \bmod M$), Modular Multiplicative Inverse ($A^{-1} \bmod M$), Euclidean GCD, and Linear Congruence solver ($Ax \equiv B \pmod M$) for cryptography and number theory.
How to Use Modulo Calculator
Step 1
Select calculation mode (Basic Remainder, Modular Power, or Modular Inverse).
Step 2
Enter integers $A$, $B$, and Modulus $M$.
Step 3
Review computed modular remainder and step-by-step Euclidean proof.
Step 4
Click "Copy Result".
Practical Use Cases for Modulo Calculator
RSA Cryptography & Diffie-Hellman Key Exchange Math
Calculate large modular powers ($g^a \bmod p$) and modular multiplicative inverses using the Extended Euclidean Algorithm for public-key encryption.
Hash Function Design & Cyclic Clock Arithmetic
Compute modular wrap-around offsets and hash bucket array distributions ($h(k) = k \bmod N$) for software data structures.
Input & Output Examples
Calculating Modular Exponentiation (Fast Power)
Base: `7`, Exponent: `256`, Modulus: `13`
Result: `7^256 mod 13 = 9` | Step-by-Step Binary Square-and-Multiply breakdown provided
Key Features & Performance
- ✓Modes: Basic Modulo ($A \bmod B$), Modular Exponentiation ($A^B \bmod M$), Modular Inverse ($A^{-1} \bmod M$), Linear Congruence Solver ($Ax \equiv B \pmod M$).
- ✓Handles negative dividends and divisors with both Mathematical (Euclidean) and Truncated (Programming `%`) rules.
- ✓Extended Euclidean Algorithm step-by-step table viewer.
- ✓100% Client-Side BigInt math.
- ✓1-Click Copy solution & step proof.
Key Terminology & Definitions
Modular Congruence ($A \equiv B \pmod M$)
A mathematical relationship stating that $A$ and $B$ leave the exact same remainder when divided by modulus $M$ (i.e. $M$ divides $A - B$).
Modular Multiplicative Inverse ($A^{-1} \bmod M$)
An integer $x$ such that $(A \times x) \equiv 1 \pmod M$, which exists if and only if $\gcd(A, M) = 1$.