As matt has said already, start from the definition of congruence. You are confusing a residue with a least positive residue (commonly referred to as a remainder).plusunim said:howcome 10=10 mod7? I thought that you can have the same numbers on both sides only when the numbers are less than the mod.
The theorem regarding mod properties states that for any two integers a and b, and a positive integer n, the following properties hold true:
a) (a + b) mod n = (a mod n + b mod n) mod n
b) (a * b) mod n = (a mod n * b mod n) mod n
c) (a - b) mod n = (a mod n - b mod n) mod n
d) (a/b) mod n = (a * b^-1) mod n, where b^-1 is the modular multiplicative inverse of b mod n.
This theorem is important in number theory and modular arithmetic, as it allows for simplification and manipulation of expressions involving modular arithmetic. It also helps in solving equations involving modular congruences.
In cryptography, the theorem is used to encrypt and decrypt messages using modular arithmetic operations. It is also used in the generation and verification of digital signatures.
No, the theorem only holds for integers and cannot be extended to non-integer values.
Yes, the theorem does not hold true if the integers a and b are not relatively prime to n. In such cases, the properties may not hold and may result in incorrect calculations.