Middle school math (India)
The product of the HCF and LCM of two numbers is equal to the product of the two numbers. Let's use this property to solve a few questions. We will prove why this is true in another video. Created by Aanand Srinivas. Created by Aanand Srinivas.
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- why product of lcm and hcf is equal to the product of the two numbers?i didnt got the logic behind this.(3 votes)
- HCF is the largest possible common factor, so it will divide both. LCM is the smallest possible multiple, so it can be divided by both of our numbers here. Let's say our numbers are 13 and 27 and we're asked to find out the LCM and HCF. If you use the traditional method to find out the HCF by prime factorisation, you'll find that the HCF is 1. Substituting our values in the equation, you'll get the following equation: 13*27=1x, where x is our unknown value. 13*27=351, which is equal to the right hand side of our equation as the RHS was multiplied by 1. This means x=351, which is the LCM for our two numbers. Take another example. Our two numbers are "xyzz" and "xyyz". The HCF would be "xyz" and the LCM would be "xyyzz". If you multiply both the numbers, you get: "xyzz"*"xyyz"="xxyyyzzz". If you multiply the LCM and HCF, you get: "xyz"*"xyyzz"="xxyyyzzz". Both the values are the same.(9 votes)
- Hi, can you also do this by dividing 6 from 18 and 12 and finding the two uncommon numbers and then multiply the uncommon numbers to 6 to get the LCM? How can we write this down with proper statements?(1 vote)
- What if I have 2x2x7 2x3x7 and 2x5x7(1 vote)
let's see what question we have a wheel given that the head CF of 12 comma Adina's 6 find the LCM okay so I'm being given the head CF of two numbers and I have to find the LCM the first question that strikes me when I say read this question is why did you give me the HCF I already know how to find the LCM if I know both these numbers I just have to prime factorize 12 write it in its prime factorize form write 18 its prime factorize form take each the highest powers of each of those prime factors and I'll get my LCM that's definitely a way you can do this problem but because they're giving us the HCF the question is giving us a clue hey can you try the other method that you know if there is another way to solve this it's a shorter way are you aware of this there is a relationship between the HCF and the LCM think about that can you recollect it the relationship is this if you have two numbers right this is a hit CF of these two numbers right and you have to find the LCM of these two numbers the kina key term here is two numbers okay so if you have the HTF of two numbers let's say head CF and then you have the LCM of two numbers of the same two numbers of course and then if you have this other quantity which is just a product of these two numbers I'm just gonna call it the product of the two numbers two numbers so these three are three quantities right head CF you know how to find LCM you know how to find product you've always known how to find user to multiply these two then there is a relationship between these and it's like the head CF into the LCM the product of the HTF in LCM will be equal to the product of the two numbers now pause for a moment that's beautiful if you think about it why this works if you have a lot of questions around this we will talk about that in our in a later video which is notice that this is actually gonna make our problem much simpler right so if you believe this to be true it's in the book so let's believe for now that this is true H safe and LCM equal plot equals product of the two numbers then how will you solve this question you don't have to prime factorize all this now because all you'll say is oh hey chief was given to me hey CF is 6 so that's given that x LCM which is what I need to find LCM we'll be equal to the product of the two numbers and the two numbers here are 12 and 18 so 12 times 18 and with this I can find my LCM I'm gonna divide both sides by that's right 6 so that I have LCM equals I divide this side by 6 and then I divide this side by 6 I get 12 times 18 divided by 6 and we can find the answer to that 6 and 12 would be this goes like 2 times 6 is 12 so I'm dividing by 6 on the numerator and denominator and if 18 times 2 which is 36 so the LCM of these two numbers 12 and 18 is 36 but you didn't do it in the usual method that you're used to you did it this time using this new property or this new result so let's add it's like this is actually the main point of this video the main point of this video is to help you know this result that heads here finding LCM equals product of the two numbers now what I want to do is really highlight this two away okay the key term here is this tool because if this had been three numbers if you had been given the head CF of 12 18 come on something else so all three numbers in HC over 6 find the LCM you cannot do this HF and LCM will not be the product of those three numbers so 2 is really the key word here so as long as the question has only two numbers in it hence you have two numbers then this result works which must raise a question right why why is it that's the result this looks so beautiful only works for two numbers think about that I mean it's a great question to think about while you do that right one of the best ways to get comfortable thinking about these things it's just solved one or two more problems do using that so let's do that let's solve one more question where we use this property now in this question I have hitting F of two numbers is 11 and the LCM is 693 so both the hit C F and the LCM are given to me one of the numbers is 99 find the other number now how do you want to think about this question you just learned the new property so your mind might be thinking oh I can definitely use it just before you conclude that always verify hey there's two numbers so it will work if it's three or four or five it it won't work so it's two numbers so yeah we can use our property and the property the HCF multiplied by the LCM will be equal to the product of the two numbers maybe for now we can call these two numbers say n1 and n2 I just see the sounding names for the just the two numbers that we're looking for so in this question what they're doing is that they're giving us the hits a of an LCM and one of the numbers and asking us for the other number so it can pause right now because after this it's about putting the numbers right in and finding the answer to the number that we want I'm gonna do it now so hit CF is what hit save verse 11 so hit CF is xi multiplied by LCM this 693 693 one of the numbers is 9999 and the other number is what we want so now I'm just gonna call it yeah n2 maybe so how do I do this now I just gonna divide both sides of the equation by 99 so that'll give me n2 + 2 equals I'm just bringing all this here okay or actually I can keep this on this side it says if it doesn't matter it so 11 into 693 11 into 693 / 99 693 / the 99 that I divided both sides by so 99 and maybe we can go left side now so what happens here so I know that 11 and 99 are nicely divisible so this is 9 times 11 so I have this 9 and 693 it should probably be divisible because we're looking for a natural number so 69 9 goes what 79763 so 7 times you have what 69 so we have six remaining 63 of 77 so there we have it the other number is 77 I can turn it on this side also 77 and we have it so this new property as you can see is pretty useful if not for anything else in the real life at least answer many questions where we know three of these things and we're asked to find one of them so all the questions you will see in fact most of the questions you will see where will be where you are given the HCF LCM and one of the numbers or LCM and two of the numbers or head CF and two of the numbers that's pretty much the types of questions we can be asked over here so you don't have to really practice hundreds of questions here right because you know all of the problems are going to be in this format you start to find out which of the three you have and find the other one so that's that's pretty much all we have to cover in this particular property except one big question which is hey why is this probably true why why does this work can we have oh no that was a big ink smudge so why does this property work and also asking why does it all only work for two numbers like the same thing has become n into n 1 and n 203 and 24 or something this doesn't work so why is that the case and I'm gonna cover that in a separate video maybe I'll call that I hate CF an LCM product of visualize to something the clue for you though before you watch that video I want you to think about this and the clue is that just think about what the HC of really is so when you prime factorize two numbers and find the heads here what are you really doing you're picking the minimum powers of each of the prime factors and the LCM you're picking the maximum powers right so think about what you're really doing when you're doing that and then think about what the product is in the same way like when you prime factor is the two numbers and think about what the product is and when you think about it you'll be able to see oh so that's why it works in the case of two numbers and why it doesn't work in the case of higher numbers