Wilson Theorem, Remainder Theorem, Fermat's Little Theorem, Euler's Theorem, Successive Division Concept, HCF(GCD) Types

In this video I have almost finished the basics of Number System as this is the 24th video lecture of my number system module.

Types of cases in Highest Common Factor/Greatest Common Divisor -

First in this video I have taken up the concept of Highest Common Factor also known as Greatest Common Divisor and I have shown how to calculate it by different methods. The methods discussed are - Factorization Method and Division Method. Then I have explained two questions through which I have shown you the most common two types of questions that are asked in the topic of Highest Common Factor. These are basically two general cases of Highest Common Factor which I have derived in this video.

What is Successive Division ?

Then I have started explaining the concept of Successive Division.Definition of Successive Division - If a number N is successively divided by x,y and z leaving remainders r1, r2 and r3 then the quotient when N is divided by x becomes the number to be divided by y and so on, therefore when we will start creating a general expression we will start from z and r3.  Successive division is very much different from individual division. In this, when we successively divide any particular number by different numbers, then after first division the quotient obtained becomes the input number to be divided by the second divisor and this goes on. I have showed how you are going to derive the expression based on the condition given to us. Some types of questions that are asked in this topic have also been taken up by me in the video.

What is Remainder Theorem ??

The theory of Remainder Theorem says that a polynomial f(x) when divided by x-1 or x+1 then the remainder can be calculated by calculating the values of f(1) and f(-1). The detailed explanation of this theorem has been done in the video and appropriate question solving has also been taken up for it and the questions that I have taken up to make you understand remainder theorem I have taken up the same questions to explain you the concept of Euler's Theorem which will also give you an idea about how to approach questions and not follow only one theorem for solving different questions. Different questions are different and we need to know all the theorems so that while attempting the exam we adopt the shortest possible way to find out the answer.

What is Euler's Theorem for finding remainders ??

In this theorem, the base in the numerator and the denominator should be coprime to each other. After that we will find out the totient number of the denominator and then Numerator base to the power totient number of the denominator will leave a remainder of 1 when divided by the denominator. This is an interesting and a very useful theorem as it would help you in a wide variety of questions.

How to find out the last two digits of higher powers of numbers ending with 5 ??

This I have not covered in my previous video.I have already told that the last two digits of these types of numbers are always either 25 or 75. Now whether it is 25 or 75 depends on the nature of the product of x and d(whether the product is odd or even) in (x5)^abcd. If the product of unit digit of power/index and tens digit of base is even number then the last two digits are 25 and if the product becomes odd then the last two digits become 75.

What is Wilson's Theorem ??

Remainder of (p-1)! divided by p when p is a prime number is -1 or (p-1) and Remainder of (p-2)! divided by p when p is a prime number is 1 and from this more tougher questions can be solved by using the concept of negative remainders. Three questions have been solved by me in the video in order to explain you the application of Wilson's Theorem.

What is Fermat's Little Theorem ??

If the Highest Common Factor of a and p is 1 or if a and p are coprime then Remainder when a^(p-1) is divided by p becomes 1. The question detailing the concept has been discussed in the video.

So friends with this I am finishing the theory part. Below are the questions that have been discussed by me in the video.

Question 1. Find the largest number with which when 72 and 119 are divided, respective remainders 3 and 4 are left ??

Question 2. Find the largest number with which when 437, 857 and 1557 are divided, leaves the same remainder in each case.

Question 3. Find the least natural number which when successively divided by 4,5 and 6 leaves respective remainders of 3,2 and 1 ?

Question 4. A number when successively divided by 3,4 and 9 leaves respective remainders of 2,3 and 7. What will be the remainders if the smallest number of the above form is divided successively by 3,5 and 7 ?

Question 5. A number when successively divided by 4,5 and 7 leave remainders as 3,4 and 6 respectively. What will be the remainders if the smallest number of the above form is successively divided by 2,5 and 7 ??

Question 6. Remainder when 2⁸² is divided by 7 ??

Question 7. Remainder when 2⁶⁹ is divided by 9 ??

Question 8. Remainder when 2⁹⁴ is divided by 15 ??

Question 9. Remainder when 27²⁷ is divided by 100 ??

Question 10. Remainder when 47¹⁰⁰ is divided by 100 ??

Question 11. Remainder when (12¹³⁾¹⁴ is divided by 145 ??

Question 12. Remainder when 5¹⁰⁰⁰ is divided by 26 ??

Question 13. Remainder when 97! is divided by 101 ??

Question 14. Remainder when 67! is divided by 71 ??

Question 15. Remainder when 20! is divided by 23 ??

Question 16. Remainder when 20²³ is divided by 23 ??

So friends I hope you would have liked this video. Please leave your suggestions and doubts in the comment section below.