# Modular Arithmetic YouTube Lecture Handouts

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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

✍ Mayank