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Number Theory

Number Theory. 04/26/07. Tasks. Announcement of Final Example lecture Course evaluation Dispatch computer project reports. Announcement of Final. The final is hold on Tuesday, May 8 th , 8:00am, Physics lecture hall Forty multiple choice problems Closed book.

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Number Theory

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  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 8th, 8:00am, Physics lecture hall • Forty multiple choice problems • Closed book

  4. Example 1-modular exponentiation Eg.1: get 510 mod 21 Tip: just remember ab mod z=[(a mod z)(b mod z)] mod z Sol: Because 52 mod 21 =25 mod 21 =4, 510 mod 21= 45 mod 21= [(42 mod 21)(43 mod 21)] mod 21=[16*1]mod 21=16

  5. Example 2-solve ax+b mod z Eg.2: [10* 510 +11] mod 21 Tip: just remember [a+b] mod z=[(a mod z)+(b mod z)] mod z Sol: Because 510 mod 21 =16, 10*510 mod 21= 160 mod 21= 13 [10* 510 +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

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