32a-Modular-Arithmetic

32a-Modular-Arithmetic - Modular IntroductionArithmetic...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Modular Arithmetic Discrete Mathematics
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Discrete Mathematics – Modular Arithmetic 32-2 Previous Lecture Common divisors The greatest common divisor Euclidean algorithm Least common multiple
Background image of page 2
Discrete Mathematics – Modular Arithmetic 32-3 Relatively Prime Numbers a and b such that gcd(a,b) = 1 are called relatively prime How many relatively prime numbers are there? Euler’s totient function φ (n) is the number of numbers k such that 0 < k < n and n and k are relatively prime. If p is prime then every k < p is relatively prime with n. Hence, φ (p) = p – 1. Lemma . If a and b are relatively prime then φ (ab) = φ (a) φ (b) Corollary . If is the prime factorization of n, then
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Discrete Mathematics – Modular Arithmetic 32-4 Congruences In some situations we care only about the remainder of an integer when it is divided by some specified positive number. For instance, when we ask what time it will be 50 hours from now, we care only about the remainder of 50 plus the current hour divided by 24. If a and b are integers and m is a positive integer, then a is congruent to b modulo m if m divides a – b. We use the notation a b (mod m) to indicate that a is congruent to b modulo m.
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/12/2010 for the course MACM 201 taught by Professor Marnimishna during the Fall '09 term at Simon Fraser.

Page1 / 14

32a-Modular-Arithmetic - Modular IntroductionArithmetic...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online