 Hello friends, in this session we are going to continue our discussion on fundamental theorem of arithmetic and we will see how we arrive at prime factorization of any composite number. So we are doing this through a problem and the problem says express each of the following positive integers as the product of its prime factors. Okay, so two composite numbers are given one is 168 and 234 clearly both are composite without any doubt why because both are even numbers and even numbers are always composite number except the number two. Okay, so how to go about it? So we use a branch tree structure for factorization. So this is how you should do it. So let us say 168 we are trying to solve problem number one and we are going to express 168 in forms of in form of product of prime factor prime factors. So how to start? Let us say it is clearly an even number. So let us divide this by two and you represent like this. So when I take two here, so hence what will be the quotient quotient will be 84. So two times 84 is 168 like that you have to see now obviously you cannot two is the prime factor. So hence you cannot really factorize it further. So let us factorize 84 further. So clearly it is two times 42. Correct, again continuing you will see it is two times 21. So if you see what am I doing? So two into 84 is 168 I am representing it as a you know as a branch tree structure and every time I get another composite number I try to factorize it further. So two times 21 is the last thing which we got. Let us take it further up and now yes. So 21 clearly is now three times seven. Now see can I go further down? I cannot really factorize seven because this is already a prime number. So this is prime number prime number all the all the numbers which we got by factorization process here the last step the last row all our prime numbers. So two is a prime two is again a prime this two is also prime three is prime and seven is prime. So hence how do we represent 168 now? So 168 is simply two times two times two times three times seven. So just write down all the prime factors which you got. So this is how I got the fundamental theorem of arithmetic or I established fundamental theorem of arithmetic that every composite number can be expressed uniquely as product of prime factors. So this is two to the power three into three to the power one and into seven to the power one. In fact if you generalize it all the prime factors or prime numbers which are there can be used to represent 168. How do I and why do I say this is because of this. So 168 could be written as two to the power three into three to the power one into seven to the power one into what are missing let's say five is missing so I can always say five to the power zero because anything any non zero number to the power zero is one similarly 11 to the power zero then 13 to the power zero into all the primes which are there let's say pr to the power zero and so on and so forth. So all the prime factors can be used to represent any composite number. So hence we got we got the canonical form. So if you remember in the last session we saw this particular form is also called canonical form canonical representation or canonical form representation. Okay canonical representation of 168. Now let's go to the next problem and the next problem is next problem is 234 right so we have to take 234. So let me let me do the prime factorization of 234. So this is B 234. Clearly it's an even number. The best would be that you start with as big factor as possible but for convenience sake let's say two we start with two. So 117 isn't it. So if you see it is 117 now clearly 117 is divisible by three. So if you see three times 39 is 117. Now 39 again can be divided as or factored as three times 13 right. Now if you see all are all the factors which we got the last row so you can you know and just encircle that. So if you see this this is what we are looking for. So so I can represent 234. 234 is equal to two times three times three times 13 that is two times two to the power one into three to the power two into 13 to the power one. So this is the canonical form. Again canonical representation okay canonical representation. So 234 could be expressed like this. Now other part of the fundamental theorem says that it is unique. You cannot really fit in any other any other prime factor here. So neither 7 nor 5 nor 11 nor 19 nothing no none other than 2, 3 and 13 would be the would be participating in the prime factorization. We will see the proof of this fundamental theorem of arithmetic in the coming sessions and Euclid gave the initial proof which was again proven by Karlsfeldt-Goss in 19th century. So in the coming few sessions we will also look at the proof of the theorem. Thanks for watching this video.