The notation ?? ≡??(modm) works somewhat in the same way as the familiar ?? =??. a can be congruent to many numbers modulo m as the following example illustrates.

sic ideas of modular arithmetic. Applications of modular arithmetic are given to divisibility tests and . o block ciphers in cryptography. Modular arithmetic lets us carry out algebraic calculations on integers …

Basic number theory fact sheet Part II: Arithmetic modulo composites Basic stuff ealing with integers N on the order of 300 digits long, (1024 bits). Unless otherwise stated, we assume N is the product of …

The upshot is that when arithmetic is done modulo n, there are really only n different kinds of numbers to worry about, because there are only n possible remainders.

Divisibility / Factoring Idiom Modulo can be used to check if n is divisible by k Definition of divisibility is if k divides n, meaning remainder is 0 To factor a number we can divide n by any of its divisors

Modular arithmetic is the “arithmetic of remainders.” The somewhat surprising fact is that modular arithmetic obeys most of the same laws that ordinary arithmetic does. This explains, for instance, …

Key notions are divisibility and congruence modulo m . Thanks to addition and multiplication properties, modular arithmetic supports familiar algebraic manipulations such as adding and multiplying together …