[Music] [Music] my name is Ravi Prakash and welcome to numbers three so this is a third topic of numbers in second topic of numbers we had discouraged lot of concepts about factors its summation product different applications from factors of numbers lot of good problems we had discussed that difference of poverty squares you remember it's applications so many things we discuss in numbers to 20 point numbers tumors I know numbers thing he number is 3 we will discuss mostly about factorials right we discuss mostly about factorials okay so know what a factorials say to introduced in numbers of two instance what is factorial you know now so factorials is nothing but multiplication right so like what is n factorial n factorial factorial is nothing but 1 into 2 into 3 so it's multiplication taken and this is n factorial right like what is what is 5 factorial it is 1 into 2 into 3 into 4 into 5 that is 120 so factorial is nothing but a mighty plication of a number starting from 1 to itself right what is 10 7 factorial so 1 into 2 into 3 into 2 7 the result will be what 5 0 4 0 okay now what is similarly what is a 0 0 factor is 1 is only important in 0 factor is 1 right so y 0 factorial factorial is 1 I will paint this concept in fundamental principle of counting will be rate please was that with you 3 point right so factorial is a kind of arrangement also in computation combination so I explain beautifully there in front I maintain principle of counting you know right so please watch it right right now we learn it 0 factor is equal to 1 okay okay so let's scatter inner star doing fine number of factors find number of factors of 15 factorial okay so find a number of factors of 15 factor right so obviously for adding number of factors with right into birth of prime business right the product of prime business right food is 15 that well you by now by now you know that basically 1 into 2 into 3 into 4 into 5 into 6 into 7 into 8 into 9 into 10 into 11 in 212 213 214 into 15 right now let me spleen this right after this after you start to respond you can solve this kind of question within way very easily right because written factorials how to calculate height for every number every remember high 8 power of every number we were method to calculate in this factorial side which really later let me explain this concept right so 15 factorial is like 1 into 2 into 3 10 15 okay so I'm writing in terms of factorial family finding number of factors means what writing in terms of prime basis for so defined this is here so two obviously three obviously five obviously seven obviously 11 the prime number 13 so they're these many prime numbers till 30 so it is in 15,000 right now what is the highest power of 13 if you read in terms of prime this is the highest power of 13 only one what is the highest power of 11 see Omaha maintains 11 occurs in 1 ohm a 10-7 occurs here 7 occurs in two times right one seven here and one seven here so 2 times 2 7 square how many times 5 occurs here so 5 occur save three times right one in five one in five another in n another in 15 right so five is occurring three times so it is 5q o mate and three occurs here so 3 C 3 equal 0 me times so 3 occurs here 1 time 3 in 6 3 occurs one time in 9 3 occurs two time in 12 3 occurs one time in 15 3 occurs how many times so 4 3 2 6 so 3 raise to 6 ok so it becomes 3 raise to 6 and 2 occurs how many times here - so - is occurring how many times here that also you can calculate C 2 is opening one time here okay two times here then again one time here also is six one time in six okay then indeed it is up it is open in three times two cubed in ten it is occurring one time in twelve it is a three two times right in 14 it is a green one time that's it right that's it Samet end-to-end occurred here so two has occurred here how many times simply counted 1 + 2 3 3 plus 1 4 + 3 7 all the green ones right sign + 1 8 8 + 2 14 sorry 8 + 2 10 10 plus 111 so - has occurred 11 times here so this is the light break up of 15 factorial in terms of prime business right in terms of prime basis so how many factors so number of factors again that would easily okay so number of factors of 15 fact on your right foot is no factors what is the number of factors power plus 1 xi plus 1 is 12 into 7 into 4 into 3 into 2 decision after dinner 84 into 484 into 4 that is 336 okay into 12 right so 33 60 + 672 is 4 9 4 0 if no tabular take it down right but I am NOT interested in finding number of factors here what I am interested interested in this concept finding number of prime factors in a factorial - is counted 11 times 3 is counted six times five three times 7/2 times level one time N thirty one thing right okay that it okay so move to legislate so if you want to find here now finding high writing will be finding highest power finding highest power of prime factor so you prime number I can read for prime number for prime number in a factorial prime number in any factorial in a knee to your right this is a sign for factorial East exclamation mark okay so finding highest power how to find the highest power of something right see suppose the first one I take first example highest power of highest power of two in 15 factorial so in large slide we saw that it was 2 raise to 11 right it was 2 raise to 11 but how how we can do directly how we can do terribly without counting at idle manual counting at obviously you can sense that it is not possible for visit numbers lengths for 40 factorial 50 factor it is not possible so 2 is 2 element how should we do that Italy right see funda will be here 15 factorial if it is right you divide it successively by 2 right let's learn the concept you the concept here is if I write like this 2 and 15 billion now I will first take single power of 2 will divide 15 factorial how many times right so we'll divide 15 taken is 15 a single power of 2 that is first power of 2 we divide 15 factor in how many time so that is 15 how many tens right to divide seven times seven leave the remainder so divide seven times okay that means 270 or 14 so single power of 2 will occur in 1515 factory seven times right understand now second power of 2 that is 2 square so two Square will occur in 1515 factorial how many times so ticket has combined rate to factor this is 2 into 2 2 square will occur in 15 factorial how many times right what is 2 into 2 4 so 4 in economic times so where it is 2 square but where does 2nd power of 2 occurs in 15 factorial so it occurs in 1 it occurs in for then it occurs in 810 digression to and that's it right only three times then that is this so you divide this for 250 so for 2 goes 15 3 times these are 3 where the second power of 2 is existent first power of 2 already counted all the numbers like 2 in 4 in 4 0 2 2 right so 1 2 is counted in the first one and second two is counted here 2 square right then in 8 there are three three twos ready we'll come to that in 12 over there two twos right so first one is counted in first one right second is called a second one now what is the highest power of 2 to 15 factorial so high each power of 2 till 15 factors how much 3 right - 2 into 2 is 3 only so one more 2 will come so you can now do it like okay you can do it like how many times this 2 cube take in terms hold right now taking this one so 2 cube will divide 15 Pepperell comment in that case you can think like third the power of to this concentrate third power of 2 will occur how many times till 15 pepper in the third power of 2 will occur only one time right third power of 2 will occur only one thing that isn't a that's it ok so to raise and shantoriya two are others in 2 then 4 then 6 then 8 then 10 then 12 and 14 right each other comment in seven times see 1 2 3 4 5 6 7 so 37 you to 2nd power of 2 occurs how many time so 2nd power of 2 11 4 8 and 12 3 times 3 and third power of 2 will occur in how many times only in 8 so only one thing right so that's it that's it so we can just add all the positive - right 7 + 3 + 1 7 plus 3 plus 1 is answer is 11 only if you got if we counted manually in the large lengths right when we counted manually in the large that we got 11 so very beautiful concert very important concept right so I each power of 2 in 15 factorial how much it is 11 what we did first we downloaded first power of 2 then second power of 2 then third power of 2 right now now if you know the concept now it is very easy because we don't have to do this much right this is is this is equal to successive division what did you do should write simply fifteen-year okay and should successively divide 2 so 2 divides 15 7 times again to divide 7 3 times again 2 divides 3 one time 7 so once you get this concept Shortridge loop so shortcut looks so beautiful right because you have the concept whole concept know right because why it is to this - 1 by 2 square by 2 cube why we did this right everything even among too short it is so beautiful now simply what engine do write 15 here and divide to successively add all the powers so divide to do it 15 7 times to divide 7 3 times 2 divides 1 so 2 divides 3 1 9 7 right okay I hope it is clear more finished plane okay so let's do questions on this factorials okay so first question will be find the highest power of we need to find the question highest power of I 2 in 60 factorial so now you know what to do right now yeah deftly what we can do we can write 60 here and we can successively / - right so divide successively to divide 60 30 times in 2 divides 30 15 times then 2 divides 15 times to divide 7 3 times 2 divides 3 one time right simply add all this presumption third e + 1545 75253 but this 156 rate it is 56 therefore highest power of 2 in 60 factorial is 1 2 raise to 56 ok highest power of 2 in 60 factor is 1 2 into 56 rate similarly did we write so we write down you okay in 60 factorial it is 2 raise to 56 okay similarly let's calculate height all of 360 controller okay so second part will be highest power of 3 in 60 factorial okay this is what so again same thing we will do we write 3 we write sit here and we'll divide successively right so 3 divides 60 20 times 3 divides 26 times G 3 divide 6 2 times so 20 plus 6 plus 2 it is 28 therefore 3 raised to 28 right so 60 factorial is what to raise to 56 into 3 raise to 28 like this right so it consists of 56 twos and twenty eight threes we mean to say right so like this you can calculate for any highest power fine okay now this is I told you this baby we are calculating only highest power of prime factors right they are targeting only highest power of prime factors okay white wine or wine or other numbers right if I write okay let me write put is highest power of 6 in 60 factorial right what is the highest power of 6 in 65 torial again you think of what is the highest power of 6 and 60 factorial still no question 3 okay this will even question 3 what is highest power of 6 and 60 factorial so I it directly can't do highest power of 6 and 60 factorial or deadly turn this if I rate 65 to earlier if I write 6 here why did that at leat under successive divide it is because 6 is not a prime number right so it can be broken into some form like 6 is also made up of 2 & 3 6 is also made up of 1 & 6 right if they are calculating for all these 6 here so 2 & 3 will be missed because 6 can be made up of 2 & 3 also right every 2 & 3 will every pair of 2 & 3 will make one things so if I calculate only 6 like 6 into 1 over a 6 7 8 so some 6 will be missed we have water calculating highest power of 6 and 60 factory right so some 6 will limit there because some 6 will be formed by this 2 into 3 also that's why we cannot calculate highest power of any non prime number by this method right we cannot calculate highest power of any non prime number by this method fine so what to do if I have at highest power of 6 and 60 factor what to do right so simply what we need to do is convert the any number into its prime factor right what is 6 6 is basically equal to 2 into 3 6 depends on 2 & 3 every pair of 2 & 3 will mix 1 6 right every one pair of 2 & 3 will make 1 6 fine so here let's say so 60 factorial is there so what is the highest power of it serial already gala radio what is the highest power of 2 in 55 60 factorial the 16 factorial consists of 2 raise to 56 and 3 raise to 28 admins there are 56 to 0 and there are 28 3 0 what is the highest power of 6 what is the highest power of 6 60 factorial consists of 2 raised to 50 6 into 3 raise to 28 so high each fall of sexual water it's like same thing right how many pairs of 2 & 3 I can make so whichever is listed that many pairs I can make right since there are only 28 this available I can be only 28 pairs of two three and one pair of two and three we give you one six right so there are only 28 per side it's like suppose have in right for right shoes we have 30 shoes for us right leg here 30 shoes right and four left leg you're 25 shoes how many pairs of shoes you can make only 25 right so whichever is less than that is a determining factor so here also if there are 56 to serve a label and if there are 28 these available how many pairs we can make I can be only 28 days ride from the rest 28,000 right from the Reg 28 tools I can't can't make any six that is a waste for me right reg 28 tools for this question is a wedge right so I can make only six raise to 28 so I get power of 6 will be 1 it is 6 H to 28 because 2 into 3 raise to 28 is the highest I can form fine this is the highest I can form okay I hope it is clear right so for any non prime number I need to break into pine numbers why because for a non find over there multiple ways to meet that number like suppose it was six there are multiple ways like 1 into 6 or 2 into 3 is 12 right so the up suppose I need to calculate create power of 12 in 60 factorial right so again if you calculate well directly 12 is made up of 6 and 2 is also 12 is made up of 4 and 3 is also write 4 into 3 12 6 into 2 is 12 1 into 12 is 12 itself right so there are multiple ways so some will be meshed if you do it directly right so that's why what is the way they should be we should make we should break into some prime factor and then we should a solve it right so see one interesting pattern here actually if I calculate number of five sources you write what is the highest power of five in 65 problem let me download it right what is the highest power of five in 60 factorial the fourth question so what is the highest power of five and City factorial now quickly target in your mind high eighth power of five in sixty Factory simply thing like this safety is our number here five will divide successively so 5 divided 60 12 times 5 divides well 2 times our intent right how much 14 so it is like 14 therefore high 8 power of 5 in 60 factor is what 5 is 2 folding so it is let's take one more into 5 into 4 be correct so you can see here obviously that 2 is the smallest number right - is this constant factor so power of 2 is 60 factorial is 56 fine power of 3 in 60 factor is 28 right it is obviously less than number of tools power of 5 in 60 fabulous 14 further design number of threes right that is why because more these prime factors will increase right more the number will be lesser the power will be right so like if I make again for 7 or what is the power of 7 so obviously power of 7 and sixty factorial will be less than will be less than 14 was right you can actually see this less than photo desires 407 and 60 buck torreón do it quickly so I each power of 7 in 60 factorial okay do it quickly highest power of 7 in 60 factorial do it mentally this question 5 it minute so 7 divided 16 8 times 7 divide 60 so 7 divides 8 one thing is 9 only right but is 7 raised to 9 for them so see powers kept on decreasing 56 then 28 and 14 then 9 right powers kept on decreasing that basically means that more is the prime number value lesser is the power right so from next time if I need to calculate highest power of 6 is 60 factorial right so I don't need to calculate both the number of 2 and 3 right even if 6 is made up of 2 and 3 6 is bit o of 2 and 3 I don't need to calculate 2 and 3 why because I know that always power of 2 will be I know that always power of 2 will be more than power of 3 okay so obviously I know that number of tools in any right number of tools in any factorial number of tools will be greater than or equal to number of threes in some cases can be equal to awesome right is get anybody to number of fives right is greater than equal to number of sevens right and so on this will go on right infinity more is a prime value lesser is the number okay so number of two's is readable number of three right so if six is made up of two and three so that I can find what is like what is the highest power of six for calculating okay so we can note note a point here for calculating calculating very point point eight very important point for calculating highest power by each power of six in 60 factorial we just need to we just need to calculate calculate highest power of three we just need to calculate highest power of three right we don't need the number of two's right we don't need number of tools number of to this highest purpose rate I don't need because I know that there will be more tools and less threes right so high each power of six is what 828 it simply depends on power of three to twenty eight right does it have to do a thing with numbers number of to 0 right so I can always make it a rule like in case of variable it is like single power rate obviously it is not double power it right and not in case of two square into 3 right not in case of of two square into 32 cube into three square right so there I cannot say which is greater but if it is single power then for sure for sure more is the magnitude less is the power okay and the final answer will depend on what only that higher magnitude one okay example in it okay liquid way to move initial aid liquid version six zero what is highest power of highest power of let's say ten in 65 to notice highest power of 10 in 60 factorial right now see again 10 is a non prime number directly I can't use it so what is 10 10 is basically two into five in terms of prime factors it is 1 2 into 5 so obviously I can not think that number of twos in 60 factorial in 60 factorial number of tools number of clothes will be much greater than number of files right so how many pairs of tools how many pairs of 10 I can mean so so I how many pairs of two and five I can be rate so one pair of two and five will give you 110 correct so like suppose there are 100 twos and there are suppose for example there are 100 tools and there are only 25 sites so how many pair I can make so I can make only 20 pairs right I can make only 20 pairs this 100 is of no use to me right I can make only 20 pairs okay so all this this will become onward this will depend on what is it depend on the higher magnitude prime factor higher magnitude prime factor is 5 right simply I can say here simply I can say here and that very point concept of trailing zeros right finding trailing zeros right so slowly I'll come to that so highest power of 10 I eat power of 10 is equal to highest power of 5 in last slide highest power of 6 was equal to highest power of 3 because 6 is 2 into 3 3 will be more 2 will be less right so it depends on gasps oh I'm sorry a highest power of 6 equal 2 into 3 3 will be less 2 will be moving right so it depends on 3 only here 10 is 1 2 into 5 so here obviously in 60 factor 2 will be more 5 will be less so Miliband only on five right so one pair of two and five will give you 110 so highest power of 10 is equal to high eight power of five cut it so what is eyes for tenure so you simply calculate simply divide five by sixty factorial rate so 5 goes to air time again 5 goes to times so five is immittance fourteen times right 14 times so I H power of five is equal to what 14 then fourth that means highest power of 10 is 14 highest power of 10 is 14 in 60 factorial right that simply depends on number of fives right now see is power of 10 is 14 it's fine but this is this one more concept here since there is 10 raise to 14 so there will be 14 zeros right 10 squared is 100 two zeros 10 squares 100 two zeroes 10 cube is thousand three zeroes 10 raise to 4 will have 4 0 similarly 10 raise to 14 will have 14 zeros so I can say in 60 factorial in 60 factorial 10 raise to 14 basically means that there are 14 trailing zeros right 14 trailing zeros trailing zeros means want trailing zeros means when a number is written how many zeros are there any left right so 6 divided at 65 is a big value right so very fine numbers 2 5 2 3 5 7 11 13 with many powers right but at last whatever is the value right whatever is the value right let's say this all numerical value inside the bracket outside there will be 14 zeros 1 2 3 4 5 6 7 8 9 10 11 12 13 14 so number will look like this right so in 60 factorial some new little value like 1 2 3 something like that anything and they are like 14 telling 0 0 14 trailing zeros right so now I can say that if you have to find number of trailing zeros that is simply equal to what it is simply equal to finding number of types because number of trailing zeros because now were of trailing zeros number of drilling zeros is equal to number of tenths and number of tenths is equal to number of files right number of files so calculating number of trailing zeros is directly equal to what number of fives so if you have to find number of trailing zeros in any factorial as simplified number of five slide okay for example question seven number of trailing zeros number of trailing zeros in hundred factorial is is Compton right simply Auto very low number of trailing zeros is equal to number of files right number of files that's it right do it mentally now number of files right I'm a number of fives 5 divides hundred twenty times 584 times we do it mentally 20 plus 4 24 that's it right so number of X is what five days to 24 simply I do it do this in my knowledge I don't know don't need to write it unites hundred fact earlier 100 right 5 divides hundred 20 times 5 divides 24 times 20 plus 4 24 that separate wood middle okay so number of fives is what 5 raised to 24 this is the answer so that 5 is 225 is what Tara therefore at 24 trailing zeros 24 trailing zeros in 100 factorial right now on trailing through the super concept right will get you to tail right after 2-3 coaches will again come back to it and we'll get into detail detailing of trailing zeros right one more person in turn iterative highest power of highest power of 8 in 150 factorial right I need to find highest power of eight in 150 factory right how you do it so high H power of 18 one 55.88 is known on a to the non prime number what is it basically it is nothing but two - okay so kate is nothing but too funny so I need to calculate number of to this area rate so I need to calculate number of tools okay alternate number of tools do it quickly right 150 here or 155 - whatever we store a trade so it is number is 152 divides 150 75 times to divide 75 37 times to divide 57 18 times 2 divides 89 times then 2 divides 9 4 times then 2 divides 4 two times in one thing right so basically you add all this what is the number 75 + 37 1 1 2 plus 18 130 139 147 148 say it is how much 112 131 39 143 145 146 2 raise to 146 if not done any calculation will take it should be fine right okay 2 raise to 146 so in 140 factorial it is 2 raise to 146 right but what I want I want highest power of 8 right what is is power of 80 mm - to write simply 2 cube that means bridge to cube here right so 2 Q by 1 so what is 146 divided by 3 so 1 46 divided by 3 is what 15 - 3 is 1 450 right so 48 into 3 is 144 so I can write okay 2 cube into 3 into 48 is 144 right and 2 square is left out okay - square is left out but this 2 square is a waste for me right it is of no use for me because this 2 square won't form any aid right because there are no tools left in 155 tray remember up to this what 146 so 146 so for 8 for 8 what I want how many triplets of 2 into 2 into 2 I can meet right that means how - cube I can move so how many Triplets of tools you can think there are 146 tools right so there are 146 in twos in terms of you how many groups of 3 3 you can mean that every groups of 3 2 will give you 1 8 2 into 2 into 2 will give you 1 8 right so how many groups of CCO can make 146 divided by 3 so 48 groups of 3 3 you can meet right so like 3 into 48 is what 144 so you can say what is answer here should be what 8 raised to 48 that's it it's 2 squared is of no use to me so 8 raised to 48 so high is power of 8 in 150 factorial is what a today's to 48 right okay next question [Music]