# Modular Arithmetic YouTube Lecture Handouts

Dr. Manishika Jain- Join online Paper 1 intensive course. Includes tests and expected questions.

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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- 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