CSC2110 Discrete MathematicsTutorial 5

GCD and Modular Arithmetic

Hackson Leung

Agenda

• Greatest Common Divisor– Euclid’s Algorithm– Extended Euclid’s Algorithm

• Modular Arithmetic– Basic Manipulations– Multiplicative Inverse– Fermat’s Little Theorem– Wilson’s Theorem

Number Theory

• Throughout the whole tutorial, we assume, unless otherwise specified, that all variables are integers

Euclid’s Algorithm

• Main idea:

• So we iteratively do divisions

• And is gcd of and

Euclid’s Algorithm

• Example 1– Find gcd(2110, 1130)

Euclid’s Algorithm

• Example 2– Given two sticks

– By elongating the sticks with same length, find the smallest positive difference in length between the two stick piles

Length = 2020

Length = 2100

Euclid’s Algorithm

• Example 2– Observation: We want to minimize positive z

such that

– Hint: spc(a, b) = gcd(a, b)– Extension 1: If we allow z to be non-negative,

• Can z be even smaller?• Shortest length of stick piles, respectively?

Extended Euclid’s Algorithm

• Example 2 (Extension 2)– I want to know how many sticks of each of two

lengths so that z > 0 is minimized– Things on hand:– Want to know:

Extended Euclid’s Algorithm

• Key: Trace from the steps of Euclid’s algorithm

• gcd(2100, 2020) = 20

Extended Euclid’s Algorithm

• Key: Trace from the steps of Euclid’s algorithm

Modular Arithmetic

• Know what it means, first!

• Which means

• Which means– a and b have same remainder when divided

by n

Basic Manipulations

• Given

•

•

•

•

•

Basic Manipulations

• Examples

Basic Manipulations

• Example– Using modular arithmetic, prove that a

positive integer N is divisible by 3 if and only if sum of digits is divisible by 3

Basic Manipulations

• We can express N in the following way

• We can say

• Since , hence

• Conclusion:

Multiplicative Inverse

• Definition:– We say A’ is the multiplicative inverse of A

modulo N

• Theorem:– A’ exists if and only if– We also say that A and N are co-prime– Note: N is NOT necessarily prime

Multiplicative Inverse

• Example– Find the multiplicative inverse of 211 modulo

101

Fermat’s Little Theorem

• If p is prime and a is not multiple of p, then

• Example 1: Calculate– Are 2110 and 1009 co-prime? – If so, by the theorem,– By multiplication rule, – Same as finding– Ans:

Fermat’s Little Theorem

• Example 2– Show that, if p is prime and co-prime with a,

the multiplicative inverse of a modulo p, denoted by , has the same remainder as

when divided by p.– Observation

• By the theorem and multiplication rule, we can say

Fermat’s Little Theorem

• Example 2 (Cont’d)– Observation

• By the theorem and multiplication rule, we can say

• Then,

Wilson’s Theorem

• It states that

• What if p is not prime?– p = 4, trivial– p > 5,

Wilson’s Theorem

• What if p is prime?– Remember the proof of Fermat’s Little Theorem?– shows a permutation of – Write them down in the yth column of a table– Each row and column has exactly a single 1– Pair up and it becomes– Only for y = 1 and y = p-1,– So,

The End