The Chinese Remainder Theorem

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The Chinese Remainder Theorem

• Solve
• x 2mod3
• x 1mod4
• x 3mod5
• Tell me the characteristics of x.

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• x 2mod3
• x 1mod4
• x 3mod5
• M 34560
• M1 60/3 20 M2 M3

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• M1 20 m1 3
• Solve 20x11 mod3
• 2x11 mod3
• x1

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• M2 15 m2 4
• Solve 15x21 mod4
• 3x21 mod4
• x2

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• M3 12 m3 5
• Solve 12x31 mod5
• 2x31 mod5
• x3

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• X b1M1x1b2M2x2b3M3x3
• 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

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Example to Try

• Solve
• x2 mod3
• x3mod5
• x2mod7
• Solution x23323mod105

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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.)