The Chinese Remainder Theorem

Sep 04, 2012

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The Chinese Remainder Theorem

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1. The Chinese Remainder Theorem Solve x = 2mod3 x = 1mod4 x = 3mod5 Tell me the characteristics of x.

4. x = 2mod3 x = 1mod4 x = 3mod5 M = 3�4�5=60 M1 = 60/3 = 20 M2 = M3=

5. M1 = 20 m1 = 3 Solve 20x1=1 mod3 2x1=1 mod3 x1 =

6. M2 = 15 m2 = 4 Solve 15x2=1 mod4 3x2=1 mod4 x2 =

7. M3 = 12 m3 = 5 Solve 12x3=1 mod5 2x3=1 mod5 x3 =

8. X = b1M1x1+b2M2x2+b3M3x3 =2(20)(2)+1(15)(3)+3(12)(3)=233mod60 = 53mod60 So any positive integer congruent to 53 mod 60 is a solution to the system

9. Example to Try Solve x=2 mod3 x=3mod5 x=2mod7 Solution x=233=23mod105

10. Homework If time permits, look at exercise 35 on page 58. Read section 2.3 Exercises 33 ace f(to hand in), 35 (note that all mi are relatively prime. You can show that some of the congruences are redundant and can therefore be eliminated.)

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Presentation Transcript

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