Modular Arithmetic

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Modular Arithmetic

• Lee Cerval-Peña

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Background

• Started with taking elementary mathematics from
  an advanced standpoint
• Decided to go in opposite direction!
• Break advanced topic into its elementary
  components
• Chose modular arithmetic (MA)

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Basic Outline

• Aimed at Year 9
• Try to teach MA up to ax ? b (mod m)
• Break down into bite-size chunks
• All necessary knowledge learnt by Year 9
• Chinese Remainder Theorem Year 11 knowledge
  preferable

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Main Points

• Introduce basic concepts of MA
• Go through rules/properties
• Do a ? x (mod m)
• Teach Euclidean Algorithm
• Do ax ? b (mod m)
• Conclude with summary and CRT teaser

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Introduce basic concepts

• Basics come by considering its other name,
  clock arithmetic
• Use visual aids to try and get idea across
• Consider various clocks with different number of
  hours in the day as examples
• Get them to understand
• 1?a (mod m) ? 1asome multiple of m

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Example worksheet question
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Go through rules/properties

• Only give basic few, dont overload them
• Brief outline of proof visual rather than
  mathematical
• Do examples of each with class

i) If a?b (mod m) then kakb (mod m), for
k?Z ii) If a?b (mod m) b?c (mod m) then a?c
(mod m) iii) If a?b (mod m) a'?b' (mod
m) aa'?bb' (mod m) aa'?bb' (mod m) iv) If
a?b (mod m) d?m, then a?b (mod m) v) If a?b
(mod m) a?b (mod m'), then a?b (mod mm')
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Do a ? x (mod m)

• Fairly straight forward
• Same as done on clock handout
• Introduce more formal notation
• E.g. Solve the following
• 13 ? __ (mod 9)
• 18 ? __ (mod 9)
• Variety of examples

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Teach Euclidean Algorithm

• Motivate topic needed for ax ? b (mod m)
• Can, again, be broken into easy portions
• Review concept of remainder in division
• Get them to practice giving integer part of
  division and remainder term
• Formalise method to give EA
• Use handout to work through examples

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Example question

• Solve 617 8 r 5
• Remember you can write this as
• 61 7? 8 5
• and 7 5? 1 2
• and 5 2? 1 1
• and 2 1? 1 0
• So gcd(61,7) 1
• Are 61 and 7 coprime? Yes

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Do ax ? b (mod m)

• Go through method very slowly, ensuring
  understanding
• Go through a number of simple to hard
  examples
• Get class to attempt further questions
• Answer all questions as clearly as possible

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Conclusion

• Use handout with brief summary of topic
• Include example questions/solutions
• Mention where they might meet it
• Also mention simultaneous equations and how this
  can lead to the Chinese Remainder Theorem

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Summary

• Theoretical plan at best!
• Good way to use time productively
• Even better way of revising topics
• Provide look forward to degree maths
• Pupils can enjoy learning the topic without
  stress of having to memorise it!

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THE END

• Any Questions?