LCM visualized

now if you asked me to find the LCM of two numbers using prime factorization I know what to do but the question that remains in my mind is do I truly understand what I'm doing to get the answer like I'm finding the prime factors taking the maximum power is multiplying them all together do I can I explain it in simple words to myself it feels like there's something to understand here and I know that the best way to understand anything is to really ask as many why and how questions that we can come up with even if they sound like dumb questions in fact especially if they sound like dumb questions and get to the bottom of what we're really trying to understand so let's do that with LCM today let's try and look at the LCM with a new pair of eyes so let's say two small numbers say 12 and 18 and ask if I have to find the LCM of these two numbers using prime factorization what's tip number one for us the step number one is to go to 12 and ask hey 12 what are your prime factors and then go to 18 and ask hey teen what are your prime factors now I did that and they told me 12 told me that it has two twos and one 3 and 18 told me that it has one two and two threes now is the time to stop and think about how to use this information to arrive at our LCM and in a previous video we did the same thing for the head CF may be visualized in a diagram y thinking about prime factors makes it easier to find the HCF that's not a strictly needed video but you may find this video much more fun if you watch that video as well so with that let's jump in and ask how to find the LCM with this information that we have right now and the first question to ask is what is the LCM really what is it isn't the LCM the first multiple of both 12 and 18 that's pretty much it right in other words if I start at the number line at 12 and keeps jumping size 12 and started 18 and keeps jumping size 18 at some point these two will have to meet and the first place where they meet is what we call the LCM so then now that we know this let's ask what should the LCM be the else 'i'm just needs to contain enough factors of both 12 and 18 so that it's a multiple of both of them think about that so what LCM is goes to 12 and asks hey 12 I'm trying to be or LCM I'm auditioning to be or LCM so what do I need how many twos do you have and 12 says so I have to do and it seems is okay I definitely need two twos then only then will I be a multiple of yours if I have only one two or zero two is I can never be a multiple of you forget being a multiple of 18 so I have two twos know how many threes do you have and Torres's oh I have one three and LCM says okay cool I'll have one three as well and then it asks 12 anything else and well says no I have nothing else so LCM is now happy I have at least covered 12 and I have not I don't yet know about 18 but I definitely know that I'm your multiple now let me go to 18 and find out the Newseum goes to 18 and asks hey woody you need I'm trying to be or LCM what do I need to have to be able to cover you and 18 says oh me I have one too the LCM looks and Sciences that's great I in fact I have two twos so no worries are all I have an extra two even though you don't need it and then LCM asks what else do you need an 18 says I need two threes to be my multiple if you have to be my multiple you need two threes at least and the LCM says oh I have only one three that twelve wanted me to have okay I can add another three so now I have two threes as well and the LCM is 18 anything else and LCM says no no Phi's with me no seven is nothing else CLC was happy the LCM says now I have the exact minimum things that I need to have to be the LCM of 12 and 18 and if you find the answer do this you will know what the LCM of these two numbers is right and you guaranteed that this number will be a cm which is a common multiple of both 12 and 18 because 223 is here already that's 12 and 2 into 3 into 3 is here that's 18 but what you also know is you have not included anything unnecessarily so you know this will be or else 'i'm right so to reduce 4 into 3 into 3 is 9 so 36 is your else 'i'm now let's visualize this in a different way now the way i like to do it is to imagine 12 to be a circle like this containing its prime factors and 18 as well like this now how is the LCM connected in this picture how should you think of finding the LCM using the picture if I were to do it what I'll do is that hey I'm still the LCM I'll go ask 12 what are your factors I need to be your multiple first if I have to be a multiple of both of you I first need to be at least your multiple and I could have picked 12 or 18 first I'm just picking 12 first so 12 LCM says okay I need to contain everything that you you contain so 2 into 2 & 2 3 that will make me your multiple great now I have another job I also need to be the multiple of 18 so I'll go to 18 and a squaddie you need an 18 says I need 2 into 3 into 3 but before adding these also to the LCM bucket the LCM realizes wait wait it's possible that some things that you need 18 were already covered by 12 so I have to ask 12 and you what's common between the two of you because whatever is common I've already covered so that's what the LCM does it wants to find out what's common and the way to do that is to just observe what's common between the two you'll notice that there's 1 2 + 1 3 that's common and because that's the case this 2 3 that you see for 18 has already been covered by becoming a multiple of 12 and that's beautiful if you think about it that part is already covered so L see you just needs to cover this extra 3 and eating will also be covered it doesn't have to cover those two and three again because it was covered for 12 now in that way when you look at it the LCM is simply this 2 multiplied by the 2 into 3 that's common which you see over here multiplied by the 3 that's not common between 12 and 18 there's an extra 3 in 18 that's not covered if you become a multiple of 12 so you only need to multiply that and now I have a question and the question is can you look at this diagram and also figure out what the head CF is notice that just by drawing the building blocks of 12 and the building blocks of 18 and bringing them together bringing them together so that the 2 and 3 of 18 and the 2 and 3 of 12 in this case those are the common factors bringing them together like this gives us both the head CF and the LCM in one diagram beautifully because once you have done this the head C is simply the common factors multiplied with each other we went really deep into this in in the previous video so 3 into 2 will be the head CF which is 6 and whatever is visible once you do this multiplied together will be or LCM 2 into 203 which is over here into 3 and now you also know why this is true I find this really beautiful let's do one more example to really understand this even better so let's take all of these out and let's take our two new numbers 120 and 84 just like we had for head CF and pause right now and think about the find the prime factors of 120 draw a diagram for it prime factors of 84 draw the diagram for it observe what's common and notice that when you merge them whatever you can see multiplied together will be your LCM I'm going to do it right now for 120 I have three twos and 1 3 and 1 5 so I have 2 cubed or three twos the short form of writing that is 2 cubed right into 3 into 5 and 484 I have two twos 1 3 and 1 7 equals 2 2 is 1 which is 2 square into 3 into 7 now finding the LCM simply becomes including all the things already included in one number so if I write the LCM out of here the LCM that'll simply be equal to 2 cube into 3 into 5 which is basically just all the numbers that 120 needs to cubed into 3 into 5 and then all the things that 84 needs that were not already covered by 120 and to find that we just find what's common and more it's the diagram whatever was what does what's part of it 84 that's already covered two twos and one three so that's already covered which means the only thing left to be covered is another 7 so you multiply by that 7 as well and then you have your LCM so let's find out what the LCM is it's 75 so 35 into 3 is 105 105 + 2 8 that's 840 840 I hope that's correct so the LCM is all the things that are visible once you merge it so that you're not counting the common factors twice because you don't need to I remember the head SIA was actually also available in this diagram directly for us the NCAA is just all these multiplied together which is two hundred one two three which is 12 so everything that we need and now is a good time to pause and ask hey how is this diagram and this way of thinking about it connect it to the metals that I'm already used to now one of the methods I was used to was writing the LCM as looking at each of the prime factors say to first and noticing what's the maximum power and then I'll write that so the maximum power in this case is 2 cubed and then go to the next one 3 3 power 1 here 3 power 1 here so into 3 5 power one here no 5 year that can be taken as 5 power 0 so taking 5 power 1 and then no 7 nothing else over here but there's one 7 here and no 7 here so taking the maximum power of that which is 1 7 notice it's exactly the same thing that you have done here and the reason you're taking the maximum power is that if one person needs to the three twos and the other person needs only two by taking the maximum you're ensuring that everybody will be happy even if there are more and more numbers as long as you have taken the maximum power of a prime factor you know everybody else will be satisfied so it makes sense to do this there's another method you may have been used to which is this way of writing things like underlined write and draw a line keep going have you done this so if you have then what you're doing is actually first finding whatever is coming 2 into 2 to 3 because you keep dividing by both the numbers and if you stop right here you get the head CF 200 203 right and then what do you start doing if you're finding the LCM is that you start looking at whatever is divisible by even one number so this in this case is to only goes in ten so you write that and then you keep finding out whatever is not common so tell you here is your what I can call your common part after which begins your not common part and observe how that neatly ties in this diagram - 2 - n - 3 is over here you multiply that alone you get your head CF there's 2 into 5 that's left behind on the 120 side and there's another 7 that is left behind on the 84 side and you multiply all of these together you were told that you'd find the LCM but now you know exactly why