1. CSC2110 Discrete Mathematics Tutorial 5 GCD and Modular Arithmetic
Hackson Leung

2. Agenda Greatest Common Divisor Modular Arithmetic Euclid’s Algorithm
Extended Euclid’s Algorithm
Modular Arithmetic
Basic Manipulations
Multiplicative Inverse
Fermat’s Little Theorem
Wilson’s Theorem

3. Number Theory
Throughout the whole tutorial, we assume, unless otherwise specified, that all variables are integers

4. Euclid’s Algorithm Main idea: So we iteratively do divisions
And is gcd of and

5. Euclid’s Algorithm
Example 1
Find gcd(2110, 1130)

6. 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

7. 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?

8. 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:

9. Extended Euclid’s Algorithm
Key: Trace from the steps of Euclid’s algorithm
gcd(2100, 2020) = 20

10. Extended Euclid’s Algorithm
Key: Trace from the steps of Euclid’s algorithm

11. Modular Arithmetic Know what it means, first! Which means
a and b have same remainder when divided by n

12. Basic Manipulations
Given

13. Basic Manipulations
Examples

14. 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

15. Basic Manipulations We can express N in the following way We can say
Since , hence
Conclusion:

16. 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

17. Multiplicative Inverse
Example
Find the multiplicative inverse of 211 modulo 101

18. 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:

19. 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

20. Fermat’s Little Theorem
Example 2 (Cont’d)
Observation
By the theorem and multiplication rule, we can say
Then,

21. Wilson’s Theorem It states that What if p is not prime? p = 4, trivial

22. 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,

23. The End