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.

We have seen that modular arithmetic can both be easier than normal arithmetic (in how powers behave), and more difficult (in that we can’t always divide). But when n is a prime number, then …

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 …

Quite surprisingly, modular-arithmetic number systems have further mathematical structure in the form of multiplicative inverses . Very roughly, sets are the mathematicians’ data structures. Informally, we …

Very basic number theory fact sheet Part I: Arithmetic modulo primes Basic stu We are dealing with primes p on the order of 300 digits long, (1024 bits). me p et Zp = 1; 2; : : : ; p p 1g. and multiplied …

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

We will soon prove a general theorem about the powers xn modulo prime numbers (such as 7) which will imply that 0; 1; and 1 are the only possible cubes modulo p = 7..