Modular Arithmetic Youtube Lecture Handouts

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Watch video lecture on YouTube: Congruence (Modular Arithmetic) & 5 Properties Explained with 7 Problems: Ultimate Shortcuts Congruence (Modular Arithmetic) & 5 Properties Explained with 7 Problems: Ultimate Shortcuts
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Modular Arithmetic

Simply Looking at the Face of Clock

Why Bother? - Shortcuts to Several Problems

  • Remainder Problems (Simple)

    • LCM

    • Chinese Remainder Theorem

  • Remainders of Exponentiations: ?

  • Last Digit Problems:

    • Modular Arithmetic

    • Euler’s and Fermat’s Little Theorem

    • Wilsons Theorem

  • More Motivations – Reducing Big Numbers

  • Time Problems

    A train coming at 3 pm is delayed 16 hours, what time will it come?

Face of a Clock

Numbers in Clock World- Concept of Congruence

Face of a Clock Replace 12 with 0- Modulo 12

What Happens with 7 Days?

Running the Clock Backwards

Addition and Subtraction of Congruence's

Application of Addition- Example-1

  • Find last digit of: 2403 + 791 + 688 + 4339

  • Remainder of

Multiplication in Congruence's

Application of Multiplication- Example-2/3

  • Find the remainder of

  • There are 44 boxes of chocolates with 113 chocolates in each box. If you sell the chocolates by dozens, how many will be leftover?

Exponentiation in Congruence's

Application of Exponentiation Example – 4/5

  • Find the last digit of .

  • Find the r

Division of Congruence's: Never Divide, Think from Basics

  • – Divide by 2

  • ( 5 and 2 are coprime) - Divide by 2

Combining Congruence's

Example - 6

3 professors begin courses of lectures on Monday, Tuesday, Wednesday and announce their intentions of lecturing at intervals of 2, 3, 4 days respectively. If there are no lectures on Saturday, after how many days will all professors omit a lecture together?

Concept of Multiplicative Inverse

  • b is multiplicative inverse of a mod N

  • a is multiplicative inverse of b mod N

Summary

  • Don’t do division without writing out basic equation

Next - Faster Solutions to Exponent Problems

  • Find the remainder

  • Euler and Fermat’s Little Theorem

  • Wilsons Theorem