 I have next question coming up for you and here it goes on your screen. In how many ways can the number one zero eight zero zero can be resolved as the product of two factors? Okay, before you start solving this one, please let me conclude the previous theory. Okay, so guys just excuse me and I'll just conclude the previous concept. So I'll generalize this. I have not generalized this yet. So kindly allow me to generalize this. Then we'll go back to that question. So let's generalize this concept. So if you have a number which is prime factorized as P1 to the par Alpha 1 P2 DT or 2 P3 to the par Okay then we say that the sum of all the devices is given by sum of all the devices is given by. p1 to the power 0, p1 to the power 1 all the way till you reach p1 to the power alpha 1 times p2 to the power 0, p2 to the power 1 all the way till p2 to the power alpha 2 and this trend continues till pk to the power 0, pk to the power 1 all the way till pk to the power alpha k. those who would be interested in knowing its simplification its simplification is p1 to the power alpha1 plus 1 minus 1 by p1 minus 1 you can note down this result as well it will save a lot of time for you later on p2 to the power alpha2 plus 1 minus 1 by p2 minus 1 okay and this multiplication continues till we reach the last term over here which is pk to the power of alpha k plus 1 minus 1 by pk minus 1 okay this result don't be surprised this comes from your GP comes from the sum of a geometric progression which anyways you are going to learn in the chapter series sequence and progressions later on with me okay now we are ready to go to the next concept next concept in fact next question that we had given over here in how many ways can you write 100800 as a product of two factors now what is the meaning of that product of two factors means let's say one way would be to write it like 108 into 100 so these are the two factors let me tell you 100 into 108 is also counted as the same way of writing it okay so don't doubly count this they are the same way of representing the same thing okay let's see somebody has replied to it yeah sure Arpita what I did okay let me just go back to the previous board once again see we had already discussed the previous example that you are going to start with the prime factor from power 0 all the way till you reach alpha 1 yes or no okay so that's what I am doing by generalizing it so remember I had a problem 24 right 24 was 2 to the power 3 into 3 to the power 1 right to find the sum what did we do 2 to the power 0 2 to the power 1 2 to the power 2 2 to the power 3 whole multiplied with 3 to the power 0 3 to the power 1 right correct so in the same way I am now generalizing it I am making it scalable so if a factor is going yeah if a factor is having alpha 1 power will start from 0 all the way till alpha 1 and keep doing this and multiply them all does it make sense to you Arpita yes sir okay good good to know that let's move on to the problem yes anybody who is ready with the answer how many ways can you write this number as product of two factors ananya no that's wrong again start small start small you will come to know the pattern no ranganath that is also not correct that is also not correct no no no no no no arpita how no shawmek no trippan okay no aditya okay we will come back to this problem after having seen the example of 24 okay so let's say I want to know in how many ways can 24 be written as a product of two factors okay now let's try to count the number of ways 1 into 24 is one way correct 2 into 12 is another way correct 3 into 8 is the third way correct 4 into 6 right next you would feel like writing 6 into 4 but remember 6 into 4 is already covered under 4 into 6 correct next you would feel like writing 8 into 3 8 into 3 is already covered under 3 into 8 next you will feel like writing 12 into 2 that is only covered under 2 into 12 correct next you will feel like writing 24 into 1 that is already covered under 1 into 24 so the answer here is you have just 4 ways okay does this ring a bell in your mind srishti you are correct awesome you were quick to pounce upon it my simple example very good srishti others did it ring a bell to you that what is happening here sir the number of ways to represent it as product of two factors is half the number total number of factors okay now who is this by the way Anurag sir Anurag okay Anurag you have concluded it very well but remember there are some exceptions to this for example let's say 36 first of all let us write down all the devices of 36 okay what are the devices of 36 can we all correct Krishna you are correct okay all of you please pay attention here 36 has factors 1 2 3 4 6 9 12 18 36 right can you count how many of these are there 1 2 3 4 5 6 7 8 9 now if you say Anurag your answer for writing 36 as a product of two factors is 9 by 2 doesn't it become absurd because 9 by 2 is a fraction 9 by 2 is a fraction right so what went wrong okay that worked for this one but did not work for this Anurag can you tell me why it didn't work for this anybody open to anybody why that half the number of factors did not work for 36 36 is a perfect square absolutely so when a perfect square comes in that approach of halfing the total number of factors of a number to find the number of ways in which you can express as a product of two factors fails right why because I'll show you right now 1 into 36 correct it has another factor 36 into 1 that's why there's a pairing happening here okay so I'll write it down over here so this pairs up with 36 into 1 okay there's a pairing happening between these two then there's 2 into 18 then there's something called 18 into 2 so there's a pairing happening between these two fellows then there is 3 into 12 12 into 3 so there's a pairing happening between these two correct then there's 4 into 9 then there's 9 into 4 so there's a pairing happening between these two now when there's 6 into 6 there is no pair to it okay so when you see this basically what you're doing you are writing every factor at least once over here right you see it okay but here you are repeating the same factor 6 with itself okay so the number of ways in which you are going to write 36 as a product of two factors is just going to be 1 2 3 4 5 okay now how can we generalize such a thing when you have a perfect square coming up and when you have not a perfect square coming up for not a perfect square I'm sure you would have found the trick when n is not a perfect square not a perfect square so what you're going to do is you're just going to do the total number of factors the total number of factors divided by 2 this is the number of ways of writing it as a product of two factors product of two factors right but if n happens to be a perfect square it's absolutely correct Shomik, Kirtan, Siddharth you're all giving me the right one yeah so if it is the perfect square okay the number of ways would be total number of factors okay you can say minus 1 by 2 plus 1 which is actually a stupid way of saying total factors plus 1 by 2 both are same thing actually okay why I subtracted a 1 is because that repeated factor I subtracted other factors are always repeated okay so that 6 and 6 2 factors which are okay that 6 factor is a repeated factor to make a 36 that I subtracted rest others will be paired up so I divided by 2 and ultimately I added up a 1 2 account for that 6 this itself is simplified to total number of factors plus 1 by 2 so just remember this that would be sufficient so the number of ways of this is the total number of ways of writing a number as a product of two factors if the number happens to be a perfect square is that clear so now answering that previous question of mine so one okay let me go back to the previous question so page number 4 it was yes yeah so coming back to this question 1 0 8 0 0 I'm sure you would have prime factorize it by now it is 2 to the power 4 3 to the power 3 into 5 to the power 2 okay so total number of factors a total number of devices for this will be 4 plus 1 okay 3 plus 1 2 plus 1 now my question here to all of you is how do I know whether it's a perfect square or not perfect square will give an odd number of factor that's one thing another thing is that if a number is a perfect square you would realize that all the powers of these prime numbers would be even don't you think that a perfect square will have all the exponents of those prime numbers as even okay so here 3 doesn't have an even power it has a power of odd number right so it is not a perfect square it is not a perfect square if it is not a perfect square as Krishna also lightly pointed out the total number of factors will come out to be odd for a number which is a perfect square so here my number of ways in which you can write it as a product of two factors would be nothing but this divided by 2 so answer is 5 into 4 into 3 divided by 2 2 2 gone answer is 30 the right answer is that is that fine everybody are you able to connect with this concept okay so let me ask you another one please stop me if you have not understood anywhere and please write clear keep writing this clear even though if I forget to say it please write it down okay here comes the next question in front of you find the number of devices of this number thankfully this number has already been given to me as a prime factorise form which are perfect square find the number of devices of this number which are perfect square by the way it's not completely fine factorise they have tricked you this is 9 to the power 11 so be careful okay let's see who answers this first time starts now I'll give you 3 minutes to do this please reply privately to me Sharmik no that's not correct no Krishna that is also not correct no that's way too much you have to tell those factors which are perfect square themselves getting my point for example 4 could be a perfect square factor for this right no Shristi no Anurag that's not the right answer no Tirpan that's also wrong see this is just tweaking of the same concept okay I'll just tweak the same concept so Dhartha no Shankin no okay guys good to see you all working it out no Anjali that's not correct okay now all of you please listen to me see if you want a factor to be a perfect square okay remember that factor should be made up of even number of prime numbers for example 4 okay 4 is made up of 2 to the power 2 and all the other factors would be 0 0 0 right by the way this number has not been written in a proper way okay just to confuse you they have written 9 over here so what I'll do is first I will make myself are no pratham that's not correct first of all I will write this as 2 to the power 3 3 to the power 5 5 to the power 7 7 to the power 9 and again this is 3 to the power 22 yes or no right so all together I have 2 to the power 3 3 to the power 27 5 to the power 7 7 to the power 9 as our number correct okay now see if you want to make a divisor okay let's say there's a divisor called K okay this divisor whatever it is made up of let's say 2 3 5 7 it should always have even number of powers on top it should always have even even even even do you all agree with me or not yes or no yes or no just type yes shankin my god that's correct wow awesome shankin has given the right answer well done shankin okay so yes so you all of you agree with me that there has to be an even choice of power right very good now 2 to the power 3 how many even powers I can pull out I can either pull out 2 to the power 0 or 2 to the or 2 to the power 2 only correct yes or no from this guy yes or no that means only 2 ways yes or no 2 to the power 1 sorry sorry that's that's 2 to the power 0 and 2 to the power 1 no no even even powers even power is what is the even power that you can go to yes 0 and 2 only right yes no problem now 3 to the power 27 you can pull out 3 to the power 0 3 to the power 2 3 to the power 4 3 to the power 6 3 to the power 8 oh my god 10 12 14 16 18 20 22 24 26 oh my god how many are there 1 2 3 4 5 6 7 8 9 10 11 12 13 14 so 14 ways you can select the powers of 3 okay now 5 let's check 0 2 4 6 4 ways to do it okay 7 will be 0 2 4 some people have started answering it also no no no no no no Anurag Silya wrong no Krishna that is also wrong 6 and 8 that is how many 5 I guess yes or no so the total number of ways you can end up making a even device sorry perfect square divisor is nothing but the product of these numbers 2 into 14 into 4 into 5 that's nothing but 28 into 20 answer is 560 ways guys remember how used to make the factors how used to count the total factors either you could put a factor 0 number of times or one number of times or two number of times till that maximum amount you can take right so that is to give you alpha 1 plus 1 but here that same concept is slightly tweaked right that's why J doesn't want its candidate or aspirants to come up memorizing a formula you have to know the logic from which that particular formula is evolving so here I have only the liberty to pick up 2 to the power 0 or 2 to the power 2 that means only 2 options are there when I talk about picking up various powers of 2 similarly they're just 14 options when it comes to picking up you know even powers of 3 there are only 4 options for picking up even powers of 5 and again 5 options for picking up even powers of 7 so altogether 5 6 0 is the total number of ways in which you can pick up even devices from this number is that clear please type clear