Modular Multiplication Table Generator
Explore a·b (mod n), units (invertible elements), and inverse pairs with interactive visualization
Inputs
Legend
Click a header number to select it; we'll show its inverse (if it exists) and highlight cells where the product ≡ 1.
Selected Element
Units (invertible elements)
How To Use
Generate and explore modular multiplication tables with interactive features
Set Modulus
Enter the modulus n (≥ 2) for your multiplication table
Choose Display
Select full table or units-only view
Explore Patterns
Use toggles to highlight units, inverse pairs, and value patterns
Interact
Click headers to see inverses and export results
What is Modular Arithmetic?
Modular arithmetic, also known as clock arithmetic, is a system of arithmetic for integers where numbers 'wrap around' after reaching a certain value (the modulus). In modular multiplication, we compute a·b (mod n) by finding the remainder when a×b is divided by n.
Key Concepts
Units (or invertible elements) are numbers that have multiplicative inverses modulo n. A number a is a unit modulo n if and only if gcd(a,n) = 1. These elements form a group under multiplication and are crucial in many cryptographic applications.
Inverse pairs are pairs of numbers (a,b) such that a·b ≡ 1 (mod n). When you click on a unit element, the tool highlights all cells where the product equals 1, showing you the inverse relationships visually.
Applications
Modular arithmetic has numerous applications: RSA encryption relies on modular exponentiation, hash functions use modular operations, and error-correcting codes employ modular arithmetic for data integrity.
Important Notes:
- Units are elements coprime to n (gcd(a,n) = 1)
- Only units have multiplicative inverses modulo n
- The heatmap reveals patterns in the multiplication table
- Click table headers to explore specific elements and their properties
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