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
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 Example1

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

Remainder of
Multiplication in Congruence's
Application of Multiplication Example2/3

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
Mayank