1. Number Theory 04/26/07

2. Tasks Announcement of Final Example lecture Course evaluation Dispatch computer project reports

3. Announcement of Final The final is hold on Tuesday, May 8 th, 8:00am, Physics lecture hall Forty multiple choice problems Closed book

4. Example 1-modular exponentiation Eg.1: get 5 10 mod 21 Tip: just remember ab mod z=[(a mod z)(b mod z)] mod z Sol: Because 5 2 mod 21 =25 mod 21 =4, 5 10 mod 21= 4 5 mod 21= [(4 2 mod 21)(4 3 mod 21)] mod 21=[16*1]mod 21=16

5. Example 2-solve ax+b mod z Eg.2: [10* 5 10 +11] mod 21 Tip: just remember [a+b] mod z=[(a mod z)+(b mod z)] mod z Sol: Because 5 10 mod 21 =16, 10*5 10 mod 21= 160 mod 21= 13 [10* 5 10 +11] mod 21=[13+11]mod 21=3

6. Example 3-gcd Eg.3: gcd(2091,4807) Tip: just remember gcd(a,b)=gcd(b,r) Sol: gcd(2091,4807) = gcd(2091,625) = gcd(625,216) = gcd(216,193) = gcd(193,23) = gcd(23,9) =1 = gcd(9,5) = gcd(5,4) = gcd(4,1) = gcd(1,0)=1

7. Example 4-inverse mod n Eg.4: We know gcd(3,4)=1, find inverse of 3 mod 4 Tip: Use Euclidean algorithm to find gcd(a,b)=sa+tb, and s mod b. Sol: gcd(3,4)=gcd(3,1)=gcd(1,0)=1 1=3-1*2; 1=4-1*3=4-(3-1*2)*3=4-3, so s=-1 s mod b=-1 mod 4=3

8. Course Evaluation& dispatch reports 14:332:202:01