welcome to a quantitative aptitude video on HCS and LCM from career acecomm HCF and LCM is one of the bases of quantitative attitudes and hence every placement test Bank MBA and all of the exams always have two to three sums on HTML NCM HTML cm concepts are useful in other topics too and a good command over them will help you increase your score today we will see some tips and tricks to identify and solve some related to ATF and SEM very very quickly more practice questions are available on career.com where you can practice with thousands of aptitude questions and practice tests so let's begin h CF means highest common factor what is the meaning of factor fate let us take a number six six can be written as 6 into 1 it can also be written as 2 into 360 and also be written as 3 into 2 so these numbers which are there these are factors of 6 what does that mean these numbers can easily and completely divide 6 that is the meaning of factor and say if they ask you what is the highest form factor that means there are some factors okay and out of those whatever is the highest number that is the highest common factor of 6 6 is divisible by 1 2 3 as well as 6 itself so here which is highest number it is 6 so 6 is the highest common factor of 6 right now they will give you 2 or 3 numbers and out of that what they say for these two to three numbers find the highest common factor so you will find the factors for all the 3 numbers right and out of that whichever is the greatest factor that will be your h CF h CF is also known as G C D demain greatest common divisors so this it is nothing but greatest common divisor that is famous HCF greatest is highest common factor right now let us take an example and see how to calculate a CF of numbers there are various ways but the best and the easiest way is to the easiest way would be taught by us today over here okay so pay attention it is very very easy even if you see it once or twice you will be able to solve it henceforth it would it is extremely easy let us take an example let us take an example of say forty to fifty four and thirty six right now find the HCF or the highest common factor of 40 to 54 836 now whenever we are simplifying whenever we have a fraction like say 24 divided by 18 what do we do we start simplifying it and what do we get at the end to expose our six threes are so we have 6 upon 6 upon 3 4 upon 3 right so this is the reduced form here we simply divide same way we start dividing over here and simplifying now 42 54 and 36 we can see it is divisible by 2 let us write 2 and let us divide by 2 to ones are two twos are four to one 0 so 2 into 21 is 42 two twos are four 14 2010 2 into 27 is 54 2 ones are 2 16 to AIDS are 16 so these numbers are divisible by 2 now next now these numbers 21 27 and meeting these are not divisible by 2 right so let's type try 3 these are divisible by 3 now don't divide by 1 why because everything is divisible by 1 if you divide by 1 we will always get the same thing 142 54:36 so start with 2 after two let's try 3 okay three sevens are 21 three nines are 27 3 sixes are 18 okay now 7-6 they are not divisible by any number now 9 and 6 is divisible by 3 but 7 is divisible by three so all the three numbers there is no such number which is divisible which can divide all these three numbers so we have to stop over here what do we have on the left-hand side two and three right here we have seven nine and six but over here at the bottom we have seven nine and six but where they cannot be divided for the so don't consider only see the left-hand side what we have two entries so what is the HCF or various common factor for 42 54 and 56 you will find it is 2 multiplied by 3 that is 6 you can check 42 is divisible by 6 54 is divisible by 6 and 36 is divisible by 6 right there is no other number which is greater than 6 and which divides these three numbers you can take it out so highest common factor is 6 see how easy it was you just keep on dividing until you cannot divide any further okay forget the remainders and all that so just see the left-hand side and just multiply those numbers this is nothing but highest common factor all HCF this is how you find out H see if now let us see what is LCM LCM is least common multiple you know what is the multiple say we have two okay table of which two twos are four two three six two four other eight two five 10 12 14 16 18 20 all these are nothing but multiples of 2 now if you consider 8 12 16 20 okay these would be multiples of 4 in these are multiples of 2 plus these are also multiples of course but 6 10 14 these are not multiples of 4 because they do not appear in table of 4 or they are not divisible exactly by 4 so 8 12 16 20 these are exactly divisible by 4 so these are multiples of 4 right and all these are multiples of 2 now we have to see what is the exactly meaning of least common multiple let us take another number like say is 3 okay write the table of 3 3 3 2012 15 21 and so on now if we want to find out the LCM of 2 & 3 okay how to do it first find out what is LCM LCM is least common multiple so find out multiples which are common between 2 & 3 which multiples are common we have 6 okay then we left 12 over here then we left 18 which is common and so on out of that which is the least common multiple 6 is the least common multiple so 6 will be the LCM of 2 & 3 see how easy it is but every time in exam you cannot provide the tables for all the numbers and when the numbers are used it becomes very difficult to solve by this method so let us learn a very easy method to calculate the least common multiple let us take an example of say 250 100 and 125 let us find out the LCM of these three numbers okay how to find the sum of these three numbers now these three numbers are divisible by five we can see so 5 into 50 5 into 20 is 100 5 into 25 ways 1 2 3 again this is divisible by 5 keep on dividing just like we did in H C F ok what we have over here is 5 5 25 now these three numbers in it see of what we did these three numbers are not divisible by any number or any single number as such so we stopped over here but that we should not do in LCM what we will do okay these three numbers are not divisible by any single number so 3 minus 1 is 2 so two numbers are the two numbers which are divisible C 4 and 10 is divisible by 2 okay and 10 and 5 these two numbers are divisible by 5 so choose any option so let us assume that we will divide 5 n 10 by 5 5 2 's are 10 okay v 1z 5 now 4 is not divisible by 5 so write 4 as it is right again now these three numbers are not divisible by a single number right these two are divisible by 2 okay so we choose that 2 1 z 2 2 Z 2 1 z now again three numbers not divisible by a single number then two numbers right one and two or just two and one or the again it is not divisible by a single number apart from one first we tried dividing all the three numbers not possible then 3 minus 1 is true so we tried a pair of number that is two number again two numbers is not possible over here so we came down to one single number that is 2 so 2 is divisible by 2 so we like to ones are 2 2 is 1 is not divisible by 2 so write 1 as it is 1 is not divisible by 2 so write 1 as it is over here okay so we got 1 1 1 over here as remainder right so neglect these all rather even if you keep it like this it is fine what you will have over here on the left hand side 5 5 5 2 & 2 so what would be the LCM just like in a 5th we multiplied everything we will have 5 into 5 into 5 into 2 into 2 that would give us - 25 into 5 is 125 into 2 is 2 50 into 2 is 500 so LCM up to 50 100 and 125 is 500 okay see how easy it was similar to HTF only thing is we have to keep on dividing till you get 1 1 1 and then consider the multiplication on the left hand side now before going on to the sum what we will do is we'll take a look at some tips and tricks related to HCF and LCM why do we need these tips because if you are using the technique of LCM n is CF in some other topics of quantitative aptitude you cannot keep on drawing this table and all that stuff there should be another way or what we can say how we can utilize other ways to get what is the LCM or the h CS we'll see we will take a look at these tips that would make our life very easy while solving other quantitative aptitude questions now first we'll start with H si s ok let us take an example point H 3 F of 18 24 and 30 what you will do over here you will simply draw the table and then divide and find the HCF but if you observe closely what do we see all these three numbers are divisible by 2 they are divisible by 3 they're divisible by say 4 no they are not divisible by 4 they're divisible by 6 right so if we divide them by 6 what do we get we get 3 4 & 5 right 6 threes I 18 6 4 24 and 6 5.30 now they are not further divisible by 3 2 apart 3 4 & 5 these are not further divisible so the highest common factor is 6 / you see how easy without drawing the tables as we're observing we could find the answer many a times you will find numbers like 24 36 48 okay and we have to find the highest common factor for these numbers and what you will do is by observation you can see all these are nothing but multiples of 12 right apart from 12 there is no no numbers greater than 12 h CF is highest common factor so greater number should be found out apart from 12 no number greater which is greater than 12 can't divide these three numbers so it CF is 12 see once you start solving more and more sons but even by observation or looking you'll find familiar numbers and you will try combination of numbers which are familiar to you like initially when we see 24 36 48 what we will sing we will start dividing by 2 then after 2 we'll start dividing by 3 then 4 and then at the end of the day we will multiply and get the SCF s 12 but when you start seeing the numbers when you start practicing you will realize that you you are able to find out the HTM very very easily just by observation right now let's go on to how to find LCM quickly let us take an example of 12 36 72 and 144 okay so out of these numbers out of these numbers we have to find the LCM of these numbers right what we will do will draw a table and do it will find the LCM but that is not the way to go about it now take a look what we do right in LCM we have to find the least common multiple that means the number which we are going to find htm' is multiple of all these numbers right now for 12 and 36 36 is multiple of 12 that means statistics can be divided by 12 is that right don't consider 12 72 72 36 and 144 72 is a multiple of 36 right so 36 can divide 72 so do not consider 36 just consider these two numbers but when you observe closely we see that 72 is also a multiple of seventy-two into 2 is 144 that is 72 and divide 144 so don't consider 72 and just consider 144 and what is 144 over here we have to find LCM of 144 okay LCM of 144 nice LC our least common multiple of 144 is 144 into one is 144 that means 12 36 and 72 as well as 144 divide 144 very easily so least common multiple of these four numbers with 144 itself say take another example 24 36 and say 72 okay now we have to find out what is the LCM of 24 36 times 72 over here again 24 and 36 is 24 multifold divide 36 no 1024 divide 72 yes 24 into 3 is 72 so cancel 24 36 and 72 can 36 divide 72 yes 36 into 2 is 72 so it cancels what remains 72 72 so this is the SCM of these three numbers of 24 36 and 72 okay let's take another example okay let us say we have 24 we will have 36 will have 48 okay now over here 24 is multiple of 48 so don't consider 24 36 and 48 right 36 and 48 now let us take out the LCM of only these two numbers you don't need to consider so what will be the LCM over here okay we will simply divide and find the LCM right there is another way what first we calculated by this way and then violence when then we'll check out the other way which is they first divide by 12 because both the numbers are divisible by 12 12 into 336 1202 4 is 48 now we cannot divide any further okay we can have 3 over here so this will become 1 and write 4 as it is we'll have 4 over here this is not divisible by 4 so write 1 as it is and this will become 1 so we left well into 336 into 4 ok is 144 so 144 becomes the LCM of these two numbers there is the other way to calculate the same in such cases say we have statistics and we have 48 and we have to find the LCM of these two numbers take the largest number what is it 48 48 into 2 what will forget into 48 into 1 is 44 a gentle 2 is how much 96 is 96 divisible by 36 no 48 into 3 what we get 144 is 144 divisible by 36 yes so 144 becomes the LCM right but praxis this second method which I have shown right now only when the numbers are small or when they're familiar like 24 36 48 20 50 40 which are easy to multiply right and if they are difficult to multiply or if you find the numbers are big use this division method after elimination okay now there is another thing which is useful from not from the point of view of a sphere LCM sums of this topic but from the entire quantitative aptitude point of view now what happens is that sometimes many times you get fractions you will have to find one by 50 minus 1 by 45 now we generally find the same of 30 and 45 and then we do subtract over in this case how to easily find the LCM and quickly solve the question what what you need to do is 30 and 45 simply take the LCM s at t multiplied by 45 ok so assume it's 30 x 45 over here we already have 30 what is not there fortifies not there so multiply 1 by 45 minus here we already have 45 and LCM or other 13 to 45 we already have 45 we do not have 30 so multiply this one by 30 okay what do we get over here we get let's right over here we get 15 upon 30 into 45 okay 15 into 2 that would give us 1 upon 90 now this is very useful only if there is one in the numerator okay if there is some other number then the calculation would be little bit different but this is very very useful okay when there is one is in the numerator now let us take a value where one is not in the numerator check out the right hand side over here what will have say for upon 30 - say is - upon 45 okay what we will get we will have the NC ms-13 - 45 right we already have 30 over here so 45 is needed so 4 into 45 we have to do what you will get 180 minus 45 into 30 volt idea 45 you need 32 into 30 60 and we get the answer whatever is the answer okay we find the answer see how easy it gets LCM we can directly whenever there are fractions in whenever we have to solve fractions you can simply multiply the denominator and consider it as L see em now let's see some of the sums related to LCM and HCF question number one what is LCM of 36 upon 225 48 upon 150 and 72 of 165 till now we have seen the LCM of numbers like say 26 or 24 36 72 etc okay we have seen of normal numbers we have not seen the LCM and HCF of fractions so let us see how to calculate LCM and HCF all fractions okay now we want to find the LCM of 36 upon 225 48 upon 150 7200 165 this is numerator this is denominator this is numerator this is the anomaly this numerator this is denominator LCM of a fraction okay is nothing but what we have to find LCM so write LCM okay what is at the top numerators right right what is at the top numerator what is at the bottom denominators right now LCM of fraction is nothing but LCM of numerators divided by what is opposite of LCM H C F okay so in cm a fraction is nothing but LCM of numerators divided by HC m HC f of denominators remember this formula very easy nothing there is nothing over here okay now let us find out the LCM LCM of numerators what is the LCM of what are the numerator we have 56 we have 48 we have 72 let us find out that LCM how to find out the LPS all these are divisible by 5 6 ok 66.6 ages are 6 into 12 okay again all these are divisible by 2 2 threes are 2 4 2 6 . now 3 4 and 6 B 3 numbers are not divisible by a single numbers okay so 3 are not be three numbers are not divisible by single number so let us write two numbers okay which might be divisible by a formal number okay so let us consider 3 & 4 No again they are not divisible by any number let us consider 4 & 6 ok 4 & 6 is divisible by 2 so let us write two two twos are four two threes are 6 and this 3 is not divisible by two so write the three as it is right now what do we have over here again three numbers all these finding numbers are not divisible every single number okay so consider a pair of numbers say 3 & 2 no 2 & 3 again not divisible any number 3 & 3 yes it is divisible by 3 3 ones are 3-1 a 3-2 sir now what I would suggest is whenever you're solving an exam stop at this you see that there are two ones over here and you just need to write two over here you will get this one this will cover this would become 1 and this would down and one only so instead of wasting time writing all this stuff you just need to stop over here okay don't write this and what you can do it take six take two take two take three and take this two and multiply what you will get the LCM you will get it as 6 into 2 into 2 into 3 into 2 that would be 6 2 Z 12 into Z 24 24 into 2 is 48 48 of 48 into 3 is 144 144 is the inseam of numerators right now let us find HCF of denominators what are the denominator 225 150 and 65 okay again all these are divisible by 5 let us these are divisible by 5 when we divide by 5 what do we get over here five fours are 20 25 45 35 1 0 5 1513 okay is it divisible further no number ok 13 is a prime number so none of the numbers of these three numbers are not divisible by any single number so stop over here don't go for the what do we have on the left hand side just 5 so LCM of numerator is 144 what is the HC of of denominator 5 so this is nothing but the LCM of the fraction see how easy to work very very easy with practice you will be able to solve this coming hardly 30 seconds right now moving on to question number two what is H CF of 36 by 75 48 by 150 and 72 by one side event again very similar somewhat we did in the first bump okay what we have to find HPF of the fraction right what we have to find HCA right what is at the top it is always the numerator what is at the bottom it is always the denominator and HCF of fraction is a CF of numerator and what is opposite of HCA LCM see the formula is similar only you change it CF to n7 in same place here right very easy what is the H çf of the numerator water numerator 36 48 and 72 what is the highest common factor of 36 48 and 72 all these three are divisible by 12 okay there is no other number which is greater than 12 and divides all these three numbers right so all these three are divisible by 12 so hcf of numerator is 12 now let us find the LCM of denominator water denominator 75 150 135 now 75 and 150 75 can divide 150 is that right yes so don't consider 75 just consider 150 and 130 by whatever is same we get over here it would be LCM of 75 also okay let's find LCM of 150 n 135 now we see that this is divisible by 535 fifties are five twos are 25 929 okay 30 and 29 we have to find LCM 30 and 29 are they divisible by any number nope so the LCM of the denominator is CM would be 5 38:29 that would be 5 into 30 into 29 what it would be 150 into 29 now let us calculate what it would be 150 into 30 how much you will get 150 into 30 it would be four hundred four thousand five hundred minus 150 it would come out to be 4 3 5 0 this is the LCM of the denominators so what is the SCF of the fraction HCF of the numerator is 12 divided by 4 3 5 0 okay which is LCM of denominator this is the F of the fractions which are given now moving to question number three what greatest number divides 1740 to 93 and leaves remainder 4 3 and 15 respectively now another tip which would be very useful while identifying whether this term is a of SCF or LCM okay is we have to see what we have to find if we have to the greatest number we know well do we hear the words greatest we hear the word greatest in HCF highest common factor or GCD greatest common divisor so here we have to find G C D or the HC f here we don't need LCM so if they say that find the least number okay find the least number where do we hear the word least we hear it ends as cm so at that time we have to find LCM we don't need HTF over there okay but here we have HCM another name or another version where we need to find LCM is when we have to find that total of something like find the total number of marbles or find out total number of toys or something like that there also we'll need to find the LCM but over here they need a greatest number that divides okay so we need to find GCD nacl but of which numbers do we have to find GCD HCF let us see let us see a force how to solve this sum what are they given find the greatest number okay now do we know the number no let us assume that no greatest number is it what do they give that when X divides 17 that is when 17 is divided by X we get the remainder as 4 ok remainder as for what does that mean if we remove 4 from 17 okay what we will get will get 13 this 13 is exactly divisible by X ok same way whenever 42 is divisible by 8 you get the remainder as 3 that means that whenever you remove this 3 from 42 ok you will get 39 and this 39 would be exactly divisible by 3 you will get the remainder as 0 here also the remainder as 0 your remainder as 0 okay same way for 93 ok 93 when divided by X you get the remainder as 15 ok but that means that once you remove 15 form 93 what do we have you have 78 which is when divisible by this number X greatest number you will get the remain that as zero okay so we have to just remove this for three and 15 from 17 42 and 90 degrees respectively then what do we get we get 13 we get 39 and we get 78 now we can find the factors of these numbers and then we can find which is the highest common factor right so we have to find the highest common factor for 1339 and 78 we already know that 13 is a prime number okay 13 can divide 39 okay 13 can also divide 78 right so let us divide what do we get when we divide by 13 we get 13 ones are 13 threes are and 13 into 6z can we divide any further no because we are finding HCF where all the three numbers must be divided your three and six get divided by three but one does not get divided by three so stop over here don't go any further right so what do we have the HCF at 13 right so the greatest number that divides 17 42 and 93 is nothing but 13 let us check how 17 divided by 13 note you will get seven 13 ones are 13 and 17 minus 13 is 4 so 4 is the remainder 42 divided by 13 13 threes are 39 42 minus 59 is 3 so 3 is the remainder 93 divided by 13 okay 13 into 6 are 93 right 13 6 are 93 what is the remainder 93 a 13-6 of 78 93 minus 78 is 15 so 15 is the remainder see so exactly our answer is correct so 13 is the greatest number that divides these and he is the remainders did he understand what we did over here we just simply found out the divisible number we had number 17 42 and 90 degree but they were not exactly divisible they were leaving remainder so we remove the remainders from them so that we get exactly divisible numbers so that we can find the highest common factor okay moving on to the next question question number four what leads to number men / 36 24 and 16 leaves lemon as remainder in each case now er what is the word least where do we find the word list it is in LCM so we do not need HTF over here we have to find this here but LCM of which numbers do we have to fight earlier than HC of what we did we had some numbers and we found out the highest Foreman factor for those numbers after removing the remainder right but over here what we have to do we have to find the LCM of the numbers as it is right you can see 36 we have 24 and you are 16 so how to find the LCM you know okay let's start now this is division all are divisible by four four nines are four six are four fours are right we have to find LCM nine and six now all the three numbers are not divisible by a single number so nine and six let us take nine and six they are divisible by three let's write three threes are three twos are okay and here for as it is now two and four are divisible by 2 so we left to will write three as it is two ones are two twos are right now one two and three they are not divisible by any number so we will stop over here why do we stop over here because two and three again not divisible by any number one and two not divisible by any number again three N 1 not divisible by any number so what do you have a 4 LCM 4 3 2 3 & 2 so the LCM would be nothing but 4 into 3 multiplied by 2 again multiplied by 3 remainder multiplied by 2 remainder what do we get four threes are 12 twelve to thirty four twenty four into 3a 72 into two is 144 so the LCM of these three numbers is 144 now whatever they given when the least number is divided by 36 it is the remainder 11 now if you divide this LCM by 36 you will get perfect division that is v-0 same case 144 by 24 remainder 0 144 by 16 remainder is 0 but we want to remainder as 11 in each case so simply add the remainder 11 what will happen whatever do you get you will get 155 now try dividing what you lack 36 36 into we will have 36 into say 1 2 3 4 30 fiction to 4 is 144 and remainder is 11 155 by 24 24 into 6s or 24 into 3 7 2 24 into success 144 here it is 36 2 into 4 144 okay here it is 24 into 6 144 remainder is 11 again 150 5/16 same way in this case also you get remainder as element so whenever you are finding in cm and whenever the remainder is same in each case just find the LCM and then add the remainder okay so by that way you can solve this son very easy these are very small small tips and tricks with you if you solve one or two sons related to these you will be able to remember them very easily whenever remainder is same in each case add the remainder at the end to get the number okay moving on what leads to number when divided by 2048 and 36 leaves the remainders 1341 and 29 respectively now earlier we saw remainder was same but your remainders are different how to tackle it let's see what is the word given over here least where do we see the world least it is in LCM so find the LCM of which numbers these numbers as as it is they are given we have 20 we will have 48 and we lack 36 let us see what is LCM everything is divisible by 4 4 5 are 24 12 are four nines are okay 5 12 and 9 ok 12 and 9 is divisible by 3 so let us try 5 comes as notice three fours are three threes are now five four and three they cannot divided by any number so we'll stop over here so what we'll have for Villa three will have five will have four and we lastly what is the LCM over here 4 into 3 into 5 into 4 into 3 5 4 20 okay 20 into 3 is 60 16 to 3 is 180 180 into 4 okay is 720 so in cm of 2048 and 36 is 720 right now what are they given remainders are 1341 and 29 that means there is some number okay there is some number which is well divided by 20 you get remainder 30 when divided by 48 you get the remainder 41 and when divided by 36 you get the remainder 29 now observe very very carefully what do we see over here 20 and 30 the 20 and 13 the difference is 748 and 41 the difference is 736 and 29 difference is 7 so the difference is common over here in such cases what to do find the LCM subtract the difference okay what do we get 720 minus 7 would be 713 this is your answer see how easy it was whenever you will need to find LCM okay whenever you need to find least number find the LCM and when the remainders are different find the difference between the remainder and the numbers and you see that the difference is same 7 7 7 okay just subtract that 7 from the LCM or subtract the difference from the LCM and you will get the answer whenever the remainders are same what you do you add it right when so remainders when the remainders are different you subtract it you just subtract the difference right see how easy it was very easy to remember also moving on what least possible 4 digit numbers when divided by 12 16 18 and 20 leads 21 as remainder again this is like what we saw the remainder is same so what do we do we simply add the remainder to the LCM is that right and why do we take calcium because they have your least number but it is not a me least number they are given these possible four-digit number let us see how to tackle these kind of sums we know it is least so find the LCM of 12 16 18 and 20 right what is LCM everything divisible by 2 26.2 aids are too noisy to tendrá right again these numbers which are there out of that 6 8 and 10 are divisible by 2 2 threes are two fours are 9 comes allottees 2 fives are right 3 and 9 divisible by 3 so we will write 3 1 4 comes as it is 3 threes are okay 5 comes as it is now nothing is divisible so what we like 2 2 3 1 4 3 & 5 so what would be the LCM it would be 2 into 2 into 3 multiplied by 1 into 4 into 3 into 5 even if you don't consider 1 then also it is fine because x 1 is the same thing okay so don't consider 1 right what do we have over here two twos of 4 4 into 3 is 12 12 fours or 48:48 into 3 is 144 144 into 5 is how much 720 this is the LCM right but this is a three-digit number what we want is what we did earlier was that we found the LCM and added 21 to it right the remainder which is there right so if we add 21 you will get 741 but we want a four digit number and this is a three digit number so we cannot add 21 as we get right now now how to find out the least possible four digit numbers right 720 is divisible by 12 16 18 a 20 that is there because it is a LCM so multiples of 720 will also be divisible by 12 1689 20 right if 720 is divisible by 12 16 18 20 their multiples of 720 we'll also be divisible let us find multiples of 720 720 1.7 27 20 into 2 is 1/4 worth zero now this is a four-digit number right this is the number right four-digit number this is the least four digit number that we can find out now so one four four zero is divisible by 12 16 18 and 20 and in this we add the remainder what we get 1 4 6 1 so this is the answer this is the least possible four digit number which when divided by 12 16 18 in 20 is 21 all remainder we did the same thing what we did in the previous sum we added the remainder only thing here we try to reach a four digit number before adding the remainder we converted LCM to oh four digit number by taking out the multiple okay off it of the LCM and then we added the remainder moving to next question the ratio of two numbers is five is to six and their LCM is 480 then there is CF is now over here we are going to learn a concept which is extremely important okay related to ACF and LCM so pay very close attention it is not difficult it is extremely easy but it is extremely and very very very important now let us assume there are two numbers okay a and B let the it's EF lettuce is forming factor for a BB this and let the LCM be this then a into B that is product of a B that the both number is nothing but the product of the HCF of their HCA and their LCM so a multiplied by B is nothing but h CF of a b and x LCM of a B let us take an example let us assume there are two numbers 50 and 20 okay what is the highest common factor for these two numbers we know both are multiply mode can be divided by 5 so let us divide by 5 what do we get over here five threes are 15 fives are twenty we cannot divide further so do not consider these so HCF becomes five now let us find out NCM okay 15 and 20 what is the highest number 20 20 to 30 40 is not divisible by fifteen twenty threes or sixty sixty is divisible by 15 so LCM is sixty okay now what is the product a into B that is 15 into 20 that would be equal to 300 okay now what is the product of HCF of both and LCM it would be five into 60 that would be equal to 300 again so we saw a B is nothing but product of their HCF and there is C M same way let's use the technique over here to solve our some ratio of two numbers is five is to six since it is the ratio we cannot use the numbers directly let the common factor be K so what are the two numbers it is five K and six cakes what is their LCM it is 480 what is their SPF we don't know let's find out what is a CHCs means highest common factor we have already seen the two numbers are 5 k + 6 k okay a 5k + 6 k right what is the SPF which number can or what value can divide these two number obviously k so when we divide by K what do you get we get 5 over here we get 6 over here nothing is divisible no number can divide these two numbers so don't consider these we have only K so k becomes the common factor or the highest common factor or the HCF for a 5k + 6 K so on so this K for a CF now always remember whenever there is a ratio and whenever you take the common factor that will be nothing but the highest common factor you do not have to draw this table and all that stuff I do this table only to explain how K becomes the H C F okay now what do as we learnt product of two numbers 5 K into 6 K is equal to product of H CF and the LP and so this gets cancelled we have 30 k equal to 480 we'll have k equal to 16 now what is K K is nothing but the h CF so the h CF of the two numbers is 16 understood remember this formula product of tuna to multiplication or the product of their in cm and H C is moving to question number 8 h CF and LCM of two numbers is 8 and 96 some of those numbers is 56 then what is the sum of the reciprocals again very easy we have just seen the formulates the two numbers are a and B then product of a B is nothing but product of there s CF and NC m ok let the number 3 a and B so there are summation or some of the numbers a 56 right and we also know a B is nothing but 8 into 96 right don't multiply right now what are they asking some of the reciprocals what is the reciprocal of a 1 by a 1 by B what you will get over here we will get n same as a B you will have B plus a we already know value of a plus B that is 56 what is the value of a be 8 in to 96 right 8 sevens are 7 upon 96 this is the sum of their reciprocals okay see how easy it was moving to question number 9 what largest number will divide 4735 and 27 leaving same remainder in each case what will be the common remainder now here there is a bit of problem because earlier we saw we have to find the greatest number or the largest number and we had the remainders over there right and since we had the remainders over there we simply subtracted and since it is largest number we found the GCD or the HCF but over here we do not know the remainder also the remainder is common over here and we only have them numbers right and we have to find the largest number so what to do let us see how to find very very small and easy trick easy okay now what do we know largest word means we have to find the GCD or the HC s okay that is for sure now we have four numbers are three numbers 4735 and 27 we do not have any remainder to subtract from these so what to do simply subtract them from each other how to do 47 minus 35 what do you get - well 35 minus 27 what do you get 35-28 okay 27 - 47 in a cyclic manner this - this is - this and this - this what do you get - 20 you know or down - time what do you get 12 8 and 20 what we have to find HCF in each CF earlier we saw we do not directly find the GCD or they say we subtracted the remainders same way we have subtracted numbers from each other now finds a SPF of 12 8 and 20 what it would be all these are divisible by 4 four threes are twelve four twos are eight four fives are twenty okay now they are not divisible any further three two and five are not divisible any further so neglect them so the HCA for the highest common factor or largest number is full okay now we do not know the common remainder we have found out the largest number that will divide these and leave the same remainder how c-47 divided by four what you will have 4 into 11 44 you will have remainder at 335 divided by 4 for AIDS are 32 remainder is 3 27 divided by 4 for 6 are 24 remainder is 27 minus 24 3 so common remainder is 3 and the highest common factor or a high largest number which will divide different numbers it for see how easily we got the answer moving on question number 10 there are three equilateral triangles with sides 114 centimeters 76 centimeters and 152 centimeters what a maximum size scale can measure them exactly now let us see this is again a very very easy sub we see the word maximum size that means greatest that means we have to find the GCD or the HCF that is for sure but let us first understand what exactly they are asking there are three equilateral triangles okay side of this then then there is this and there there this okay side of this is 152 centimeters side of this is say 76 centimeters and this is one one four centimeters now they want a small scale okay they want a scale that can measure these side sides of these equilateral triangle perfectly or exactly should not happen that you keep the scale over here then again you see the scale over here again you keep it over here again you keep it over here and again you keep it over here and this much part is left that should not happen it should perfectly match over here okay and you cannot know this is not a regular scale where you will have markings in between it is like say a piece of log okay what size log piece you will need so that using that you can measure the entire length exactly like whenever in whenever we were kids we used to take a small piece of wood okay and using that use to measure how much is the length of the bigger piece of wood or how much is the length of the playground and all that stuff or whenever there is a tile on floor we take a pen and we try to measure using a pen how much is the length of the tile that is two pens is the length of the type the breadth of the tile is just one pen way of wire okay same way we have to use a scale scale okay which tends perfectly measure these right this is what they need some such sums are very very much what we say famous and very common in exams so let us see what exactly is the technique over here there is nothing whenever they are asking maximum size scale that can measure them exactly we just have to find the HCF of the given number one one four seventy six and 162 let us see what is the h CF / - over here what you will get to five thousand fourteen fifty seven two three so six sixteen thirty a to seven so 1412 76 19 all these three are multiples of 1993 are fifty seven 19 2008 19 fours are 76 okay so can be multiple can be divided any further 324 no so discard these don't consider what do we have - and 19 so the HCF or the highest common factor or the maximum size scale that we can that can measure would be 2 into 1980 38 centimeter let us check whether answer is correct or not 38 centimeter can measure 76 centimeter very easily say if this is the steel of 38 centimeter it would simply be double right 76 centimeter again over here we last 38 centimeter again we keep the scale of 38 centimeter again we keep the scale of 38 centimeter and you will get one 1/4 over here will keep scale of 38 centimeter will keep scale of 38 38 and again 38 fours are 38 into 4 is you will get 152 centimeter that is what they are asking now this can be asked in variety of our types okay which word fill triangles measuring the side of equilateral triangle just one way just keep in mind they generally ask what a maximum size scale can measure then exactly this would be the clue moving to next question if X minus 4 is the HC f of X square minus 8x plus 15 and x square minus KX minus 1 then what is the value of K now X minus 4 is HCF that means it is a factor of these two equations that means we divide both these equations perfectly so X minus 4 divides both these equations perfectly with zero remainder now this is very important please remember this if X minus 4 is a HCF or a factor of these new equations that means when we put the value of x equal to 4 both these equations would get satisfied that means both these equations would be equal to 0 right so X minus 4 again if you put it into X square minus KX plus KX minus 1 it would be 0 right since it is 0 0 we can equate them what do we get X square minus 8x plus 15 would be equal to X square minus KX minus 1 when when we have X equal to 4 so put the value of x we'll have 16 minus 32 plus 15 would be equal to 16 minus 4 K minus 1 this gets cancelled 15 plus 1 16 16 minus 32 minus 16 so K will be equal to 4 right value of K for see how easy it is just remember when X minus four is factor or HCF of these two equations that means put the value of x as 4 okay if X plus 4 is the factor put value of x s- work so that this should become actually zero so we have to put the opposite value right moving to question number 12 5 o'clock a ring automatically at intervals of 12 minutes 8 minutes 3 minutes 4 minutes and 10 minutes respectively in 8r from the moment they start how many times will they ring together now this is the most famous type of some that are asked under SPF and LCM whenever you see such kind of sum that there are intervals of clocks ringing or there is there are some signals on the road okay there are some signals on a road and the signals start or switch off at these intervals of time so find out at what time they will be on together always when they will be off together or over here there are how many times you will drink together after in 8 hours or something legate's always remember it is a sum of hdfn LCM it is very easy do not leave it for options now out of hdfn LCM which you want to find very easy let us see how to find out what we did in HTF we simply took some numbers and we divided them okay for LCM how do we find generally relative say we took an example of 2 and we write the table of 2 2 4 6 8 right then we take 3 right the table of 3 he says it and we have to fund LCM of these two numbers over here we find the common multiples that is the next instances where both of these means so always remember whenever we have to find out how many times will they ring together after the start it will always be the next instance and it will always be LCM just like an equilateral triangle okay we saw that we cannot find LCM we want the maximum that can measure their we have HCA over here it would always be LC and because we have to find the future future can only be only we found out in tables X future right now it is 2 1 Z 2 so after some time - 2 Z 2 or two twos are four 2 3 6 so next instances only ACM can give you right so we have to find LCM of 12 8 3 4 and 10 let us find ok 8 is divisible by 4 don't consider 4 12 is divisible by 3 don't consider 3 so in cm of 12 8 and 10 we have to find out let us see let us find out this is divisible by 2 6 okay 4 & 5 now this these to be free number is not divisible by any single number so these 2 divisible by 2 2 is 3 to write this as it is a spike okay 3 2 & 5 are not divisible any number so keep it as it is what we have 2 2 3 2 & 5 what is the LCM it would be 2 into 2 into 3 into 2 into 5 2 2 0 4 7 12 12 is our 24 into 5 120 minutes so the next time once they start the next time they will ring together would be they start together okay at this point just start together okay and next time they will ring it will only be after 120 minutes okay at this point what is 120 minutes to R that means if they start at 0 0 then at 2 o'clock they will ring together then after against words they will ring together so let us know in doubting HR from the moment they start how many times earrings now 0 0 they start okay after two hours varying together again after two are sharing together here it is towards are over here for Arthur over right after against to R we will drink together here six hours or work again after two are they will bring together here eight hours or so in 8 are so many times earrings 1 2 3 & 4 so they ring four times are from the moment they start see how easy it is whenever we have to find the next instance always take the LCM okay now mini attention exam you might instead of the clock ringing you might have a signal on a traffic signal there might be three to four traffic signals on a road okay and they might be switching on and off at some intervals of time and we have to find out when all of them show red or all of them show green and all that stuff just find the LC n moving to question number 13 three cyclists cycle along the circumference of a jungle they complete one round in 27 minutes 45 minutes and 63 minutes respectively since they start together when will they meet again at the starting position again this is similar to the previous term okay very famous kind of song these two we have to find the next instance when they meet and we know when we have to find the next instance it is just the LCN so this is the jungle okay the cyclists are over here one two and three all of them start together and you have to find when was the next instance when they meet together again at the starting point what are the times given for the cyclists twenty-seven minutes 45 minutes and 63 minutes and we have to find next instance that is LCM so find the LCM all these are divisible by nine nine threes are 95.9 sevens are three five and seven they are not or divisible by any number okay so we have the LCM as nothing but 9 into 3 into 5 into 7 9 into 3 into 5 into 7 What evil i-93 so 27 27 into 5 into 7 5 7.35 5 to 10 and 3 is 13 135 into 7 what would be dancer 75 is our 35 7 is 11 324 to 71.7 H nine nine hundred and forty five minutes after nine hundred and forty-five minutes all the three cyclists will meet again and that to exhaust starting position okay moving to next question Minaj wants to paste wallpaper on wall of his room the wall is 4 meters and 50 centimeters in length and 3 meters and 50 centimeters in height but one should be covered completely only by square pieces of wallpaper having same size what is the number of maximum sized wallpaper square needed to cover the walls completely now you might think this is such a big stump and how to solve these kind of sums this might be very lengthy but actually it is not it is very very easy over here again this is HCF and MCN kind of sum why we have to find maximum sized wallpaper square what does this mean greatest that means hi years that means G C D or H C F now what do we have to find the H here let us see this is the rooms 1 ok length is 4 meters 50 centimeters okay let's convert everything into centimeters so we'll have 4 50 centimeters and the height is 350 centimeter we have to cover this with square pieces of wallpaper in such a way that the square pieces perfectly fit over here it is just like measuring an equilateral triangle right okay you cannot have half a piece like this or something like that you should have perfect pieces over here right so we have to send what we have to see what is the maximum size of means I can fit over here so what to do just mine is er HC f of what 450 and 350 right what is it divided by 50 what units minds are sevens are so you'll have this and this and this as your a things which remain these two are not divisible okay by anything so don't consider these what is remaining this so this is the maximum size wallpaper needed HCF is 50 centimeters right now this is the maximum sized wallpaper 50 centimeter by 50 centimeters it is a square right square wallpaper pieces are needed 2 T by 60 they want what is the number of squares needed how to find number of square very very easy what is the area of the wall 450 into 350 what is the area of one square wallpaper 15 250 how many wallpapers are needed simply area of the wall divided by area of one square wallpaper right whenever we have one wallpaper and how do we find how many wallpapers are needed simply find area of the wall divided by area of one square one paper okay one square wallpaper is area and you will get the total number of wallpapers what do we have this comes out to 9 this comes out to 7 this would be 9 7 63 so 63 wallpaper pieces which are square in size and having dimension 50 by 50 centimeters are we they took over the wallpaper be very easy maximum sized wallpaper just find HPA ok and number of wallpapers are nothing but area of wall divided by area of the square piece right moving on question number 15 the sum of two numbers is 156 and there HCF is 13 the number of such number pair is now this looks a little bit different or a little bit complicated and we have not seen how to handle these kind of sums but this is pretty easy this can be solved by logic okay now what do we know there are two numbers a and B okay their sum is what 156 what is their HCF it is nothing but 13 that means that these two numbers must be multiples of 13 then only can these be divided by 13 correct so let us find out what are multiples of 30 let us write the table of 1313 1026 threes are 39 okay 52 will have 65 we will have 78 we will have 91 1 0 4 1 1 7 and 130 and 143 and 156 why should we go till 156 because they said that the sum of the two numbers is 156 okay if the if we go beyond 156 let's say if we take hundred and sixty-nine now any pair which is there already their number is 169 which is greater than 156 so we cannot use it maximum amount the maximum addition should be 156 so max we can go to 156 okay now let us see what to do what they say is sum of the two numbers is 156 now find the pair whose sum is 156 13 if added with 156 can we get 156 no not at all 13 when added with 143 can we get 156 yes make this a pair okay this is one pair now this pair 13 and 143 what is there HCM or the highest common factor we know 13 divide this so 13 is nothing but the highest common factor so we found one pair let us write over here we found one pair which is having some 156 and HCF is 13 now 13 cannot be combined with any other number to get 156 we will always get less than 156 okay 26 26 when you add with 143 okay will you add with 143 no it will go beyond 156 just remember 26 add it with 130 you will get 156 you will observe that if you want 156 you just have to go this way right one number from here and one number from here right again one number from here one number from here right then one number from here and one number from here now let us see what do we have over here okay 156 okay 26 that can be added with 130 and we'll get 156 but what is the HCF 126 and 130 for this pair this is 26 and 130 the HCF is nothing but 26 figure 26 into five is 130 so we cannot have this pair cancel it out okay next is 39 and 117 okay 117 and 39 the addition is 156 right but what is the HCF over here it is 39 because 39 into 3 is 100 17 Soyuz form-factor is 39 right it is not 13 so this pair also get cancelled out now what do we have we have 62 and one zero four right addition is 156 what is the HTF 52 into 2 is 1 0 4 so highest form factor is 52 so again it is not 13 highest common factor is not 13 so this does not satisfy our condition now we are 65 and 100 9 91 edition is 156 correct now we have 65 we have 91 edition 156 what is your HCF or the common factor 13 fives are okay and here we will have 13 in 2/7 there is no other factor to this okay no other number divides 65 and 91 only 13 divides these both so what we will have will have 13 as the HCF so this pair also satisfies condition so we have 2 pairs now how many pegs we have to pair now then we have 78 ok 78 plus 78 how much it is it is 156 and 78 again divides itself so highest common factor when it comes to 78 it is 78 only so we cannot consider this get cancel so there are only two pairs which has some 156 and hdfs 13 right moving to next question question number 16 what is the least number which when divided by number 3 5 6 8 10 in 12 leaves in each case our remainder - but which when divided by 13 leaves no remainder okay options given are a 3 1 2 9 6 2 1 5 6 - 1 5 8 6 let us add an option say none of the or because generally in exams we find that option right none of the above now what we have seen over here why have we given options over here because I want to show another trick which you could use in exam whenever time is a constrain and you are not able to solve using normal methods okay so what you can do you can take help of options now what has he given whenever the number is divided by 13 Italy is no remainder means the number is perfectly divisible by 13 let us see which of the following are divisible by 33 1 2 is divisible by 13 9 6 2 is 1 5 6 2 actually is not divisible by 13 and 1 5 6 is divisible by 13 now here I am telling you in short okay we are so there you have to check whether it is divisible by 13 or not by dividing it right so see is not the options of cancel it right we have a D and D now what every given least number which when divided by 3 5 6 8 they leave the remainder 2 so will first take option 1 312 this when divided by 3 5 6 8 10 and 12 it will relieve remainder 2 that means if we remove the remainder 2 will act 310 and this has to be perfectly divisible by 3 5 6 8 10 and 12 but we can easily see 310 is not divisible by 3 so 3 1 2 cannot be the answer next let us take option D 1 5 8 6 again this has to be divisible okay 1 5 8 6 when divided by 3 5 6 8 10 and 12 leaves remainder 2 so remove remainder 2 what you will get 1 5 8 and 4 now this has to be perfectly divisible by 3 5 6 8 10 and 12 but this is not divisible by 10 so this is not the answer answer is 962 okay now let us check if there are 4 options you can click 962 but if there is option called none of the above you have to check whether 962 is divisible by everything or not take 962 okay again remove the remainder what you will get 960 right this is divisible by 3 you can see this is divisible by 10 you can see by 12 by 8 again by 5 since it is divisible by 10 it is divisible by 5 since it is divisible by 12 it is divisible by 6 so 960 so the answer is 960 to see how easy h CM and LCM can be you just have to remember how to find ACF and LCM just practice little bit so that by seeing you will be able to find the LCM anger xcf okay and also remember the formula two numbers is the product of their HCF and they're ill see em okay with this we come to the end of tutorial on HCF and LCM if you like this video please give it a like and share it with your friends give your comments and suggestions below you can also mention topics on which you want videos we would be rolling out more such videos and tutorials so subscribe to our Channel and stay updated