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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 Numbers in Clock World- Concept of Congruence Numbers in Clock World- Concept of Congruence

Face of a Clock Replace 12 with 0- Modulo 12 Face of a Clock Replace 12 with 0- Modulo 12 Face of a Clock Replace 12 with 0- Modulo 12

What Happens with 7 Days? What Happens with 7 Days. Image What happens With 7 Days. Image

Running the Clock Backwards

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

• Remainder of

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?

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