[Music] [Music] yeah hi my name is Ray Prakash and this is the second class of factorials okay very number system part 3 so we started this question what is the highest power of 12 that divides 80 factorial right what is the highest power of 12 that divides 80 factorial right okay so that basically means that what is the highest power of 12 that divides 85 to really it means that what is the highest power of 12 what is the highest power of 12 that exists in 80 factorial higher fault well that exists in 85 to write this we have to find basically because whatever is the highest for up to 12 existing in 80 factorial that only will divide completely the highest power of 1295 Tony right simply we need to calculate what is the highest power of 12 that exists in 80 factoring right so now what is 12 12 is basically about to this square into 3 in terms of prime business right so let's calculate in 80 factor what is the highest power of 2 in 80 factorial what is it quickly do it in mind only right I am writing for your convenience but you do in your mind right to 480 factorial it's goes 40 times and 2 goes 20 times then 2 goes 10 times and 2 goes 5 times then 2 goes 2 times then 2 goes one time correct what is this sum sum is almost 75 77 78 so it is 78 therefore 10 to in 80 factorial it is a 2 raise to 78 in 80 factor it is 2 raise to 78 okay what is the highest power of 3 now build a good high 8 power of 3 in 80 factorial simply again 3 divides 80 to 86 times 3 divided by X 8 times 3 divides 8 2 times correct 26 plus 8 plus 2 that is 36 therefore high 8 power of 3 is 36 in a fine so now we need to find what is the highest power of 12 in a tea factory right so what is the highest power of 12 in 85 so you need to find what is the highest power of 12 so 12 consists of 2 square and 3 that means one pair of 2 is square and one pair of 3 will make one - all right so how many 2 square is it here so 2 is square exits how much 39 power of 2 square is 39 we can say and power of 3 is 136 so again it's the same thing right it's like suppose a raise to 39 and some berate to 36 how many pairs of a and B I can make only 36 pairs of a and B I can make it only 36 pairs of a and B I can make that means how many pairs of 2 squared + 3 I can make only 36 pairs of 2 is square and 3 can be red so I can make only 36 pairs of 2 square and three red red 3 2 squares are left right you take out 36 another three are left right this three are of no use because no three is left right I mean to say it 2 square is like here 36 times right I am in 36 pairs so another 3 another 3 power of 2 square is left right but 2 square needs with any number any 3 to make it 1 12 right so new threes are left so that 3 powers of 2 is square which are left is a waste for me right that is of no use okay so that means what is the highest power of 12 15 18 factorial 36 so 12 raised to 36 right this is answer this is the answer right away because the question was what is the highest power of 12 divides 80 factorial so if we found out that in 80 factorial high 8 power of 12 is 36 so obviously the highest power of 12 that is dividing 80 factor will be 36 only right that means 12 raised to 36 is the answer its what is the answer answer is 36 okay let's move to next question so question is question 2 what is the highest power of H power or power is also called this also right so don't get confused power I can write his index also right what is the highest index of eleven that exists 10,000 factorial upon 500 factorial okay so what is the highest power index of 11 index or power both meaning is same okay what is the highest index of 11 or IH power of 11 that exists in thousand factorial by 500 factor away it's very easy what we'll do is simply calculate in thousand factorial 11 occurs how many times right so in thousand factorial 11 occurs how many times simply in calculated eleven divides thousand how many times 90 times eleven divides 90 how many times eight times right so 90 plus eight is 98 so it is like 98 therefore it is 11 raise to n thousand factorial 11 is how much 98 okay now in five 5-minute factorial so 11 divides 532 how many times 11 divides 500 times forty five times right 14 five times and 11 divides 45 how many times four times so 45 plus four how much 49 okay so 11 raised to 49 here right here so yeah 11:49 okay that means in thousand factorial it is 11 raised to 98 and in 500 factorial 11 eight 249 right so this is it there obviously there are 98 elements here and there are 49 elements here right so both will cancel each other okay so what is the answer in that case so 98 and 49 okay both will cancel what is the answer answer is 11 raised to 49 right simply it is like a raise to M upon area to n is equal to e raise to M minus n you know this so what is the answer answer is 11 raised to 49 fine yeah move to question three now question three is a good question right I hope you'll try it before travel for I am solving this question right is it pauses between a tie discussion and try for five minutes that is a good question question is two four one nine factorial is equal to 5 0 4 raised to a into B okay raise to e into B where B is B is not a multiple of 7 B is not a multiple of seven okay so we need to find the value of a a is equal to what it's a very good question right so please try it this question two four one nine factor is equal to five zero four in raise to a into B B is not a multiple of seven it is given that so what is the value of a okay so you pause this video and try this question okay now I'm solving it see it says here that two four one nine factorial now is equal to 5 0 4 raised to a what is 5 0 4 if you break 5 0 4 now the key term is here is B is not a multiple of 7 okay and if you read if you break 5 0 4 in terms of 7 you will see that what is 5 0 4 it is 7 into 8 into 9 5 0 4 is what it is 7 into 8 into 9 raised to a and into B will be left fine 7 into 8 into 9 raised to a into B so that means now it is given that B is B is not a multiple of 7 B is not a multiple of 7 so B is not of the form 7 K okay so if B is not a multiple of 7 that means all these sevens on the right side let's say this is LHS and if is RHS right this whole is our ages so all the term on the right hand side right contains 7 and that should be equal to number of sevens in the left hand side obviously because it's equality sign right so number of sevens in the left side should be equal to number of sevens in the right side of asleep right and I know that B is not a multiple of seven so all the numbers all the number of sevens in the right hand side all the number of sevens in the right hand side are are concentrated in seven raise to K only so I can I can write I can write like this some 7 raised to a into 8 raised to a into 9 raised to K into B I can write R just like this right 7 raised to a into 8 raised to a into line rate to a into B this term inside the bracket this term inside the bracket doesn't contain a 7 right doesn't contain 7 okay that means all these sevens in the right hand side should be concentrated should be concentrated involved in 7 raised to K so number of sevens in LHS should be equal to let me write here number of sevens in LHS should be equal to number of sevens in our ages obviously not only number of sevens everything every if I if I break these factors into number of prime factors every term in left side should be equal to right side right number of sevens in left side should be equal to number of 7 separate side number of twos in left side should be grow number of twos in right side number of fives in left side should be grown over rights in five number of fives in right side right so every term has to be equal right but we will concentrate only on 7 because 7 is the key factor here right be it is given question that B is not a multiple of 7 P is not a multiple of seven right so we can concentrate only on 7 year because it's a two-for-one nine factorial is a huge number right you contain so many prime factor gives in a zoom rate so insensate how many prime factors one 225 prime factors their powers 100 or 200 therefore the 21 prime numbers rights to their powers like the active 2 4 1 9 factorial it's a huge number right so let's say that all are concentrated in P right all those are acquiesce B is a variable and since both side has to be equal to all are concentrated and B only every other power right so I'm just concerned with number of sevens so number means of sevens in a latest should be equal to number of sevens in our ages ad should be other numbers also right so let's find what is what are the number of sevens in a lettuce so number of seven in l a-- tests are number of seven in two four one nine factorial simply divide it to four one nine factorial number of sevens so seven goes how many times so again if you can remember it with seven in reg - 4 is 2 4 0 1 right so from there you can get so k 7 into seven cube that is 343 - 4 0 1 so 7 will go 345 times further two times right so seven goes 345 times right okay and then again 7 goes how many times with three forty five to seven who goes seven for 28 and goes 49 times because seven into 14 or 343 again 7 will go seven time again 7 will go one time simply add all these numbers what are the numbers 345 plus 49 394 + 7 4 0 1 + 1 4 0 2 so power of 7 is 4 0 2 right that means let's say on right side it is like 7 rate on left side it is like 2 4 1 9 factorial is equal to 7 raised to 4 0 2 in to let's say some key and key is a huge huge number right because the rest of the values of key will come will combine with 78240 - to form 2 4 1 9 that relates ok is a huge number but I am only concentrator is number of sevens so number of sevens in left side should be equal to number of sevens in right side because inside bracket there are no servants I'm sure about it because it is given that B is not a multiple of 7 right so what is the value of a if I equate it therefore 7 raised to a is equal to 7 raised to 4 0 2 therefore E is equal toward a is equal to four zero to a beautiful question and absolutely fabulous question right so you should be able to solve this kind of question right so get this concept very good concept okay fine yeah okay let's move to next problem now let me spin a concept here a concept of is skipping zeroes right a concept of skipping zeroes okay so next concept will discuss is of skipping zeroes right skipping zeroes now very important concept lots of good questions are based on this concept right is skipping zeroes in factorial right now what do you mean by skipping zeroes in fact all right see when a 0 is when 0 is formed right I discuss the concept of this rate trailing zeroes them we discuss the concept of trailing zeroes right so what is how a trailing zero is formed right so a trailing zero is formed by number of five sides explained this concept that it is formed by number of fives because trailing zeroes is equal to number of tens and number of tens is equal to number of two and five and always number of two will be less than number of five so in number two will be written in the greater than number of fives so number of five still it aside how many zeroes in a factorial right so number of trading zeroes is directly equal to what number of fives okay so see when a zeroes is formed like in five factorial you see it falls to zero will you form in five federal rate for scaling 0 will come in 5 factorial because 4 to 5 will come in 5 factorial for example for factorial about 4 factorial is 1 into 2 into 3 into 4 that is 24 it doesn't have a 0 right 3 factorial is 6 it doesn't have a 0 ok 5 factorial is what 24 into 5 120 what is 5 factorial 24 is 4 factorial 2 into 5 that is 120 right it'll have one tailings one trailing zero why because 1/5 is there no now in six factorial also it is 721 trailing zero seven factorial also write fifty forty so one trailing zeroes so in all these factorials there will be one trailing zero right why because there will be only one five in five factors also one five in six factor also Phi will come only one time in seven factorial also five will come only one time right so till nine factorial it will have only one trailing zeroes large will have only in 10 factorial if you calculate a value now first time it will be like some number and it will lead to trailing zero that a large white two trailing zeroes in the FATA large because 10 factorial contains two five 10 factorial contains two fives it contains two pipes right 1/10 factor is what 1 into 2 into 3 till 10 so 1 5 will come in 5 and other five will come in 10 right so there are 2 5 so there are two trailing zeroes like like this omitting 0 so I can say that from 1 factorial 2 4 factorial unshut is concept right 1 factorial 2 4 factorial 1 factorial 2 4 factorial no trailing zeroes no trailing zeroes okay 5 factorial 2 9 factorial 1 trailing 0 1 trailing 0 right 10 factorial 2 14 factorial how many zeroes two trailing zeroes right because till 14 factorial till 14 factorial 5 occurs only 2 times right so only two trailing zeroes number of ready zeroes is what equal to number of fives so 10 5 12 to 14 factorial 2 trillion zeroes right similarly 15 factor will have how many zeroes now 3 delusional you go 3 fives will be there through telling zeroes 15 factorial to 19 factorial all values will have all values will have three trailing zeroes right just watch this CD difference here 20 factor a 224 factorial again 20 factor we will have four trailing zeros and note trailing zero is added in 21 factorial 22 factorial 23 it is 24 because no 5 is multiplied so no trailing 0 will be added right now this 25 factorial right here is that fault is skipping zero that concept 25 factorial will have vomited in zeros so since twenty four factorial have four trailing zeros for trailing zeros right so 25 factor is twenty four factorial to 25 factorial 25 is multiplied what is 25 25 is 5 square so 2 more zeros are added right there are 6 trailing zeros 6 trailing zeros right c10 is the skip d-10 is skipped 1 0 is right so 24 5 12 25 factorial 24 factorio had four trailing zeros in 25 factors certainly 2 zeros are added because two pipes are added to 5 sub x so suddenly two zeros are added so 4 plus 2 is 6 so 6 reading zeros right so if I ask you a question how many numbers how many numbers exist with four trailing zeros there are five such factorial right 19 factorial sorry 20 factorial 21 factor by 2020 right this is fit for for belly zeroes how many factors exist for 5 trailing zeros for five 1000 such factorial because in 24 factors is four trillion zeros in 24 days 6 finding the right but there is no there is no such value of any factorial in which there are 5 trailing zeros right this is the skipping 0 concept okay now you can sense it when does this the skipping 0 will happen right when one 0 will be skipped right so it will be skipped when suddenly multiple powers of 5 are added red like 24 factorial will have had four trailing zeros so 25 factorial had 6 trailing zeros right so 1 trailing zeros now another went in certainly to values when it will be stable now now again 25 factorial to 29 factorial all will have six trainings it was only right because no fives are multiplied till 25 to 29 in salute 10 0 is added right again 30 factorial 1 are added so again 7 training rate is pending to go on again 30 factorial 234 factorial again there will be only 7 trailing zeros only 7 trailing zeroes right 32 34 7 trailing zeros 25 to 29 6 trailing zeros right so when a 0 will be skipped now it will be escaped when another add another multiple of 5 square right another multiple of 5 square so like 49 first multiple was 5 squares for 25 so next multiple cut it for 25 is 150 so 49 factorial will have let us say if it has X zeroes right if it has X 0 H trailing zeroes then 15 factor will have how many trailing zeros X plus 2 trailing zeros it in target exactly right how many 5s important in factorial simply calculated in mind 5 goes 9 times 5 goes one time so 9 plus 110 is there are 10 trailing zeros in forty nine factorial 10 threading zeros so in 50 factors it will tear a bit 12 trailing zeros 12 trailing zeros right in 50 factorial because 49 factorial to 50 factorial 50 is multiplied what is 50 it consists of 5 square v square means certainly two pipes are multiplied so now two zeros will be added right so 10 to 12 3 so 1 0 will be skipped at every multiple of 25 right 1 0 will be s cubed at every multiple of 25 right now just think when like this the gap of 1 0 right 10 to 12 can there be gap of 2 numbers can siddell you go from X to X plus 3 like here it is X to X plus X to X plus 2 gap of one number right gap of one number can it can it go from X to X plus 3 right that means 4 X to X plus 3 suddenly three zeros is to be added right when three zeros will be added so three zeros will be added when three fives are multiplied when three fives are multiplied five cube what is 5q 125 years so if 124 factorial will have will have X zeros right if 124 factor will have X zeros try to integrate a value of x obviously then 125 factors will have how many zeros it will have x + 3 0 zs x + 3 zeros because 124 factorial 2 125 factorial 125 is multiplied what is 125 it is 5q so now three 5s are multiplied it certainly so three zeros will be added right so X 2 X + 3 right you can always download then what is the value in 125 factorial + 125 factor what is the number of fives calculate simply five goes 25 if I was five I was 131 so in 125 factor there were 31 zeros 31 trailing zeros so obviously in 124 that well 3 less right there are 28 trailing zeros so beautiful concepts and lot of good cons questions are based on this concept okay so you got to remember this rate okay so let's do a question on this question okay so the question here will be no no factorial right no factor find II can write like this find the find the minimum value of K find the minimum value of K for which for which no factorial has no factorial has key trailing zeros K trailing zeros or T plus 1 trailing zeros or on tapeless to reduce an absolutely fantastic question right super concept this is key and many times you see in CAD is that in emissions a lot of questions are asked when this is skipping zeroes right good questions are asked so this is one of those very good question right so find the minimum value of n sorry find the minimum value of K for which no factorial has K trailing zeros or k +1 trailing zeroes or keep rotating zeros right just understanding language right what is this language is a perfect language a bit off we need to comprehend correctly to solve this question right if you get this language what is this language all about right this will be there it will solve the question because we have explained is concept earlier also okay just try and try pauses video for five minutes and just try this question right read this question at least five times this is this these are the questions which will raise your IQ level raise our intelligence level right okay so let's understand this concept right for next 5 minutes please concentrate very well now first let's comprehend the question find the minimum value of K for which no factorial has key trailing zeroes for k plus 1 trailing zeros or k plus 2 trailing zeroes right now what are this key k plus 1 and k plus 2 what does these three numbers represent these three number represents these are three consecutive numbers three consecutive numbers that means there is such any value of such a value of any factorial in that factorial they are known K trailing zeros or K plus-1 telling zeros or keepers to telling zeroes right that means three consecutive numbers are escaping R is given correct we just discussed this concept for let's say 120 4 factorial had 28 zeros right previous slide 28 zeros and 1 to 25 factorial had 31 trailing zeros right so I can tell you finally this quotient can be converted to find the minimum value of K for which new factorial has T plus 1 trailing zeros or K plus two trailing zeros because here to conquer what is K plus 1 and k plus 2 these are two conjured in the Miss rate or I can simply write it as I can simply write it as K and K plus 1 ok better idea does K and k plus 1 no factory less k trailing zeros or k plus 1 trailing zeros this question can be find a minimum value of K for which new factorial has a trailing zeros or k plus 1 2000 that means two consecutive numbers are stupid right so where it will be skipped two consecutive numbers will be escaped when a factorial has x0 and the net factorial has X plus 3 zeroes right so two numbers are a script in this case two numbers are escaped what is the answer for this question K represents 29 here and K plus 1 depends 30 years right no factorial has 29 and 30 zeros right to equality numbers are scripted right similarly this question is for three consecutive numbers being is scripted K trailing zeros k plus 1 trailing zeros and keep the two trailing zeros right so three consecutive numbers numbers are escaped so when three cons video numbers will be skipped see two consecutive numbers are escaped when suddenly three fives are multiplied so three fives are multiplied that means three zeroes are added so if three zeroes are added like 28 to 31 like 28 to 31 so two numbers are skipped coded into 13 so when similarly I have to find those three numbers to be a skeptic right then four fives needs to be added sorry for five needs to be multiplied so that four zeros are added right four fives needs to be multiplied so that four zeros are added when four fives will be multiplied when four five silla multiplied think for a minute when four fives will be multiplied what is five raise to 4 625 that's it yeah 625 you think is right okay five six twenty four factorial if it had X 0 X trailing zeros okay then 625 factorial will have obviously X plus 4 trailing zeros it will have X plus 4 trailing zeros right very good concept why because from 624 factorial to 625 factorial certainly 5h to four is multiplied so 5 raised to 400 that means four fives are multiplied if four fives are multiplied that means four zeros will be added right so you can also calculate net salary at number six 25 factorial had how many fives so five will divide 125 if I will divide again 25 again five and then one right how many zeros it has 156 so 625 factorial had 150 six zeros so obviously 624 factorial has one how many zeros it has 150 two zeroes right 4 zeros less so 624 factorial has 150 two trailing zeros 625 factorial has 156 failing zeros right how many zeros are added four zeros how many numbers are escaped so four zeroes added that means three consecutive numbers is skempton right that means what are the values of this key K president keep - - so value of this K T plus 1 and K + 2 are what value of this key K + 1 - R T is equal to 153 T plus 1 is equal to 154 and k plus 2 equal to what 155 now just read this question again this question says that find the minimum value of K for which no factorial has K trailing zeros or K plus-1 telling zeros or K + 2 trailing zeros right find the minimum value of K right that means 3 Vidia numbers doesn't exist right so no factorial has just three quantitative numbers what are the three conservative numbers what are those three consecutive numbers 153 154 and 155 fine very good so what is the meaning value of K so answer for this the minimum value of K is what 150 now wait discussion not over this co-signs over pedis concert is not over right I will come back to it right just see this if I've any doubt I don't think it will be ready it's a simple concept of zero is skipped I split beautifully in the our slide please watch that again and see k k plus one and keep - two that means certainly three cons of idiot numbers that is scripted so three consecutive numbers will be skipped only when four fives are multiplied that means when four zeros are added so three numbers will be skipped four zeroes are will be added in five event four five silly multiplied when four fives will you multiplied five raise to 4 what is 5 into 4 625 right 624 - 625 so the answer is crochets point of 2/3 right now one more thing here why this is minimum value this is important right why this is minimum value here right find the minimum value of K for which no factor has K trailing zeros or K plus 1 or K / - why this minimum value explained you see this minimum value because this minimum value because at every multiple of 6 25 factorial now four zeros will be added right four zeros are added first time at this term right 624 - 625 but again niche time also to be added right when the next multiple of 625 that is 1250 so from 12 or 49 factorial to 1250 factorial again same thing will happen if 1249 factorial has let's say a zeros a zero then 1250 factors will have a plus four zeroes we will have a plus four zeroes right because 1249 to 1250 again what is multiplied 1250 12:40 9 to 12 15 what is multiplied 1250 2002 what is 1250 625 into to what is 625 625 is what 5 raised to 4 this is 5 so again 625 is multiplied so again 4 zeros will be added right so again these are those also numbers rate again I can say the value of K k plus 1 k plus 2 his word again these three numbers are skipped but obviously the number of zeros in 1250 factorial will be much more than this 625 filter rate so that's why I need to find the minimum value so that's why answer is what answer is 153 is correct answer I needed to find a minimum value because this will this pattern will happen at every multiple of 625 625 then 1250 then 1875 then 625 into 4 is 2500 right at every multiple of 625 this pattern will happen that are multiple are number before a multiple of 625 will have a zeros and then it do that when 625 multiplied or it's multiple it will have a plus four zeros right but I have to find the minimum value so what is the minimum value it is 153 right so 153 is the answer and absolutely super question ok so just re why is it and keep this concept in mind yeah [Music]