Modular Operations
Result
9
17 + 23 ≡ 9 (mod 31)
Operation Breakdown
Raw Result
40
Quotient
1
Remainder
9
Modular Inverse of b
27
Modular Clock Visualization
Step-by-Step Solution
Step 1: Calculate 17 + 23 = 40
Step 2: Divide 40 by 31: quotient = 1, remainder = 9
Step 3: Result: 9 (mod 31)
Fast Modular Exponentiation
Modular Power Result
9
7^256 ≡ 9 (mod 13)
Binary Breakdown of Exponent
100000000
Exponentiation Steps Visualization
Square-and-Multiply Steps
Extended Euclidean Algorithm
GCD Results
GCD(a,b)
2
Bézout x
-9
Bézout y
47
Mod Inverse
N/A
Bézout's Identity
240 × (-9) + 46 × 47 = 2
Extended Euclidean Algorithm Table
| Step | Quotient | Remainder | x | y |
|---|
GCD Reduction Visualization
Linear Congruence & CRT Solver
Solution
x ≡ 6 (mod 7)
Verification: 3×6 ≡ 18 ≡ 4 (mod 7) ✓
Solution Process
Step 1: Check if gcd(3, 7) divides 4
Step 2: Find modular inverse of 3 mod 7
Step 3: Multiply both sides by inverse
Solution Set Visualization
Complete Solution Set
Cryptographic Applications
Generated Keys
Public Key (n, e)
(3233, 17)
Private Key (n, d)
(3233, 2753)
Encryption/Decryption Demo
Original Message
42
→
Encrypted
2557
→
Decrypted
42