Congruence and Congruence Classes

Proof. We first show that no two of 0 1

Solutions to Homework Set 3 (Solutions to Homework Problems

If GCD(a n) = 1

3 Congruence

Proof: If a ? b mod n then by definition n

(b?a). Theorem 3.12 If ab ? ac mod n and if gcd(a n)=1

CS 30: Discrete Math in CS (Winter 2020): Lecture 26

26-Feb-2020 and only one integer b ? Zn such that ab ?n 1. This number is called the multiplicative inverse of a modulo n and denoted as a?1. Proof.

Number Theory

the congruence class; the canonical representative of [a]n is b) ab ? cd mod n. Proof. Let a b

Math 110 Homework 1 Solutions

15-Jan-2015 Solution: (a) The integer m divides n if there exists an integer r such that n ... inverse (mod n): there exists b ? Z such that ab ? 1.

Problem Set 4 Solutions

22-Feb-2005 Prove that there exists an integer k?1 such that. k k?1 ? 1 (mod n). · provided gcd(k n)=1. Assume n > 1. Solution. If gcd(k

THE MULTIPLICATIVE GROUP (Z/nZ) Contents 1. Introduction 1 2

Then gcd(ab n)=1. Proof : By Lemma 2.3

2. To determine the gcd(24 · 32 · 5 · 7 2) 2 · 3 3 · 7 · 11) take the

Let n be a fixed positive integer greater than 1. If a mod n = a and b mod n = b prove that (a + b) mod n = a + b and (ab) mod n = (a b ). Proof:.

Math 110 Homework 3 Solutions

29-Jan-2015 (b) This allows simplifications of the computation of ab (mod n) ... Hint: You can prove it in a way very similar to our proof in lecture ...