x ≡ y mod n
x is congruent to y mod n when x modulo n gives the remainder as y
This can also be represented as x = kn + y
x ≡ y mod n if n divides (x - y)
This property can be used to verify that the first equation is correct
Properties
Given x ≡ y mod n and a ≡ b mod n
then (x + a) ≡ (y + b) mod n
Given x ≡ y mod n and a ≡ b mod n
then (x - a) ≡ (y - b) mod n
Given x ≡ y mod n and a ≡ b mod n
then (x * a) ≡ (y * b) mod n
[(a mod n) + (b mod n)] mod n ≡ (a + b) mod n
[(a mod n) - (b mod n)] mod n ≡ (a - b) mod n
[(a mod n) * (b mod n)] mod n ≡ (a * b) mod n
| Property | Expression |
|---|---|
| Commutative Laws | (a + b) mod n = (b + a) mod n(a * b) mod n = (b * a) mod n |
| Associative Laws | [(a + b) + c] mod n = [a + (b + c)] mod n [(a * b) * c] mod n = [a * (b * c)] mod n |
| Distributive Laws | [a * (b + c)] mod n = [(a * b) + (a * c) mod n] |
| Identities | (0 + a) mod n = a mod n (1 * a) mod n = a mod n |
| Additive Laws | For each a∈Zn there exists ‘-a’ such that a + (-a) ≡ 0 mod n |