1. CSE 20: Discrete Mathematics for Computer Science Prof. Shachar Lovett

2. Today’s Topics: 1. Modular arithmetics 2

3. 1. Modular Arithmetic 3

4. Modular arithmetics 4

5. Residue classes  Remember that equivalence relations = partition of universe to equivalence classes  Even = [0 mod 2] = {…,-4,-2,0,0,2,4,…}  Odd = [1 mod 2] = {…,-3,-1,1,3,…}  Partition by value modulo 2 5

6. Residue classes  Similarly, there are 3 residue classes modulo 3  [0 mod 3] = {x: x=3a} = {…,-6,-3,0,3,6,…}  [1 mod 3] = {x: x=3a+1} = {…,-5,-2,1,4,7,…}  [2 mod 3] = {x: x=3a+2} = {…,-4,-1,2,5,8,…}  Similarly, modulo n defines n residue classes. For x  Z, we denote by [x mod n] its residue class. 6

7. Modular addition  We can add residue classes.  Fix n.  [x mod n]+[y mod n]=[x+y mod n]  This is well defined. What does that mean? A. For any x,y, the sum x+y is defined B. The definition of sum does not depend on choice of representatives C. There is a way to obtain all residue classes D. I don’t know 7

8. Modular addition  Residue classes mod 5 8 0 mod 5 1 mod 5 2 mod 53 mod 5 4 mod 5

9. Modular addition  Residue classes mod 5  [x mod 5]+[1 mod 5]=[x+1 mod 5] 9 0 mod 5 1 mod 5 2 mod 53 mod 5 4 mod 5

10. Modular addition  [x mod n]+[y mod n]=[x+y mod n]  Theorem: this is well defined  This means:  if x,x’  Z are such that x mod n=x’ mod n;  and y,y’  Z are such that y mod n=y’ mod n.  Then (x+y) mod n = (x’+y’) mod n.  So, the definition of addition does not depend on the chosen representatives for each residue class 10

11. Modular addition  [x mod n]+[y mod n]=[x+y mod n]  Theorem: this is well defined  Proof (direct proof):  If x mod n=x’ mod n then x’=x+na for a  Z  If y mod n=y’ mod n then y’=y+nb for b  Z  Then (x’+y’)=(x+y)+n(a+b)  So [x’+y’ mod n]=[x+y mod n]  QED 11

12. Modular addition  [2 mod 5]+[4 mod 5]= A. 0 B. [2 mod 5] C. [1 mod 5] D. [1 mod 4] E. None/other/more than one 12

13. Modular multiplication  [x mod n]*[y mod n]=[x*y mod n]  Theorem: this is well defined  Proof (direct proof):  If x mod n=x’ mod n then x’=x+na for a  Z  If y mod n=y’ mod n then y’=y+nb for b  Z  Then (x’*y’)=(x*y)+n(ay+bx+nab)  So [x’*y’ mod n]=[x*y mod n]  QED 13

14. Modular multiplication  [2 mod 5]*[4 mod 5]= A. [1 mod 5] B. [2 mod 5] C. [3 mod 5] D. [4 mod 5] E. None/other/more than one 14

15. Modular multiplication  [2 mod 5]*[4 mod 7]= A. [1 mod 5] B. [2 mod 7] C. [3 mod 35] D. [8 mod 35] E. None/other/more than one 15

16. Modular multiplication  X={0,1,2,3,4}  Define: xRy if [2x mod 5]=[y mod 5]  Is R… A. Symmetric B. Reflexive C. Transitive D. More than one E. None 16

17. The ring Z n  Define Z n ={[0 mod n],[1 mod n],…,[n-1 mod n]}  The set of all residue classes modulo n  We defined addition and multiplication on Z n  You can check: they obey all the usual definitions of addition and multiplication (abbrv. [x]=[x mod n])  [x]+[y]=[y]+[x]; [x]*[y]=[y]*[x]  [x]*([y]+[z])=[x]*[y]+[x]*[z]  [x]+[0]=[x]; [x]*[1]=[x]  …  Sets which support addition and multiplication are called rings 17

18. The ring Z n  If n is not prime, weird stuff can happen  Example: [2 mod 6] * [3 mod 6] = A. [0 mod 6] B. [2 mod 6] C. [3 mod 6] D. [5 mod 6] E. None/other/more than one 18

19. The ring Z n  If n is not prime, weird stuff can happen  Example: [2 mod 6] * [3 mod 6] = [0 mod 6]  That is, we multiplied two nonzero elements, and got zero as a result  (these are called zero divisors) 19

20. Applications of modular arithmetic