Modular Arithmetic Properties Calculator
Test associative, commutative, and distributive properties of modular addition and multiplication
Test Modular Arithmetic Properties
Property Test Results
Associative Property Test
Addition Test
Left side: (7 + 5) + 3 ≡ 1 + 3 ≡ 4 (mod 11)
Right side: 7 + (5 + 3) ≡ 7 + 8 ≡ 4 (mod 11)
Result: ✓ Associative
Multiplication Test
Left side: (7 × 5) × 3 ≡ 2 × 3 ≡ 6 (mod 11)
Right side: 7 × (5 × 3) ≡ 7 × 4 ≡ 6 (mod 11)
Result: ✓ Associative
Example Tests
Associative Addition (mod 7)
Test: (3 + 4) + 5 vs 3 + (4 + 5)
Left: 7 + 5 ≡ 0 + 5 ≡ 5 (mod 7)
Right: 3 + 9 ≡ 3 + 2 ≡ 5 (mod 7)
Result: ✓ Associative
Distributive Property (mod 6)
Test: 2 × (3 + 4) vs (2 × 3) + (2 × 4)
Left: 2 × 7 ≡ 2 × 1 ≡ 2 (mod 6)
Right: 6 + 8 ≡ 0 + 2 ≡ 2 (mod 6)
Result: ✓ Distributive
Algebraic Properties
Associative
(a ○ b) ○ c = a ○ (b ○ c)
Order of operations doesn't matter
Commutative
a ○ b = b ○ a
Order of operands doesn't matter
Distributive
a × (b + c) = (a × b) + (a × c)
Multiplication distributes over addition
Modular Arithmetic Facts
All three properties hold for modular addition
All three properties hold for modular multiplication
Multiplication distributes over addition in modular arithmetic
These properties are fundamental to number theory
Quick Tips
Test with different values to verify properties
Modular arithmetic behaves like regular arithmetic
These properties enable algebraic manipulations
Used in cryptography and computer science
Understanding Modular Arithmetic Properties
What is Modular Arithmetic?
Modular arithmetic is a system of arithmetic for integers where numbers "wrap around" when reaching a certain value called the modulus. It's like arithmetic on a clock face.
Key Definitions
- •Modular Addition: (a + b) mod n = ((a mod n) + (b mod n)) mod n
- •Modular Multiplication: (a × b) mod n = ((a mod n) × (b mod n)) mod n
- •Congruence: a ≡ b (mod n) means n divides (a - b)
Property Proofs
Associative Property
Follows from associativity of regular arithmetic. Since (x + y) + z = x + (y + z) for integers, the same holds modulo n.
Commutative Property
Similarly follows from commutativity of regular arithmetic: x + y = y + x and x × y = y × x for integers.
Distributive Property
x × (y + z) = x × y + x × z holds for integers, so it also holds modulo n.