okay so for question 9 d uh we have 65 and 169 uh so what I'm going to show you right now is a method that I call L division that's not the official name but there's an L init so I call it l division um and our goal here is to try and divide both of these by the largest possible number that will work now in this example we can only divide 65 and 169 by a single number so this is kind of not the best example I might do maybe C or something as well after this okay but yeah does anyone know what both of these numbers can be divided yeah 13 yeah very good so again if you're not sure CU some of the numbers like again these are odds so two doesn't work right if you're not sure just check the primes so yeah we can divide both of these by 13 and then 65 13 is 5 and 169 is 13 times itself so we going be left with this can I still divide the five and the 13 by the same number here the answer is no so we have to stop here because there is nothing that works with the five and 13 the greatest common factor GCF is going to be all the numbers that are on the left side here and then multiply together okay now here we only have one number here the 13 so that is our GCF so I would just write 13 here and then you I would expect you to be doing the L division like literally on the on the page here and working your way down but I did it here just so I can make it bigger for you to see okay I'm going to do another example here I'll do c here just because the uh the numbers are I know the numbers will uh there'll be more than one number here so I'm going to go ahead and do c for you also so this is going to be 48 and 136 so again L division like this so I expect you to have a setup kind of like this what can I divide both of these but let's just pick a smallest number yeah two is the most obvious one right so let's go ahead and do that again the point here is that you can do it in multiple steps you don't have to go for the largest number right away okay so when I divide 48 by two I get 24 yeah half and when I get uh sorry when I divide 136 and half I get 68 yeah 68 very good so that's our first step right now we can keep going right and that's the whole point of this is you can do it in multiple steps right 24 and 68 can also both be divided by what two two again yeah so next layer 2 24 2 is 12 12 68 / by two is 34 can I divide 12 and 34 by the same number yes next layer two I heard again okay 12/ 2 is 6 34 ID 2 is 17 can I divide 6 and 17 by the same number no because 17 is prime right it's part of our list so we can only divide the 17 by 17 or one but dividing by one doesn't do anything because it doesn't make the numbers small and obviously 17 does not work with the six so we stop here now once we stopped now we look at all the numbers on the left side there and we multiply all of them together so this is going to be 2 * 2 * 2 which will give us8 so this one here the GCF is going to be eight get that from the two * 2 * two okay so that's question 2 D and I guess I just did C for you okay so if you have three numbers uh for example like question J here what do we do different so it's slightly different it's it's still like similar setup right I still draw the L and and I ask okay what can I divide all three of these numbers by right are there any numbers that uh will divide into all three of these two works for this but not that three three yeah I also heard someone Mumble 11 but I couldn't tell who said it but yes three and 11 both work I'm going to go three here when I divide this but that's going to be 22 I divide this by three I get it looks like 55 and then when I divide this by three looks like I get 77 and then this is where the 11 comes in right someone said 11 earlier as well so we can keep going when I divide this by 11 2 5 7 can I keep going no yeah so here we get like three prime numbers right so it's not going to work anymore and here we stop and again we take the left side numbers there the GCF in question J is going to be 33 like this okay and here's the important part the next question we're going to do in question 10 it says the least common multiple or LCM as I'll say for short the difference between GCF and LCM they're they're pretty similar but in GCF it needs to work for all the numbers in your list if it only works for for example two numbers then you can't do it you have to stop okay so yeah that's that's the important part you'll notice that when I'm doing the LCM later even if it only works for two numbers I actually can keep going yeah so LCM is kind of like this to you but GCF we stop uh once uh all the numbers cannot be divided by the same thing anymore and we only take the numbers on the left okay that's that all right so for number 10 e we're looking for the uh lowest common multiple here so this is the uh the smallest number that uh both the 30 and the 55 can fit into like multiplication wise the setup for this is very similar to The GC c f the greatest common factor that we did earlier right it's still L division um but you'll notice when we get to three numbers and four numbers it it'll be quite different even with two numbers it's a little bit different already what can I divide 30 and 55 by five y so I'm going to go ahead and do that so six and then that's going to be 11 can I keep going here no so then I would stop here now if this was GCF the GCF would be five but this is not GCF this is LCM right lowest common multiple for lowest common multiple for LCM you have to take the L shape of all the numbers that are on the outside like this instead of just the ones on the left okay for GCF you only take the left like a five here but for L CM L you take the entire L shape of everything okay and then you multiply it all together so in this case it would be 5 * 6 * 11 so you punch that into your calculator 5 * 6 is 30 and then * 11 will oops * 113 it's going to be 330 yeah and that is our LCM for question e okay so that is the difference between DCF and lcf all right now in K there's three numbers here and the process is even more different than before but first let's do the division here what can we divide all three of these one 11 good so we do that we get two here three here and six here are there any other numbers I can divide all of them by no for GCF we would stop here but for LCM we keep going but now we look for two uh numbers that we can divide by the same number and then we we continue okay so all three there's none left great next step now we look at two numbers that we can divide by the same thing so can you give me two numbers here that both can be divided by the same thing three and three and six yeah the three and six can both be divided by three so we continue three is there and you're Wonder Mr W that the two cannot be divided by the three what what do we do the answer is you just leave it alone okay the other ones that you can divide nicely you you divide those so this is going to be one and then the six of course is going and same thing we're working on two numbers at a time now right are there two numbers here that can be divided by the same thing yes right obviously the two twos can both be divided by two so I do that next 2ide two is one one we we leave that alone because one cannot be divid by two and then two at this point we now stop because we can't go any further right we cannot break ones any less now we multiply everything in the L shape now in this case because there's all ones on the bottom it the ones on the bottom don't matter right so here we just go 11 * 3 * 2 but if there were other numbers down here still that were not reduced down to one then we would have to multiply by those also we'll probably run into that in question P probably I haven't actually done it but probably so here 11 * 3 is 33 and then * 2 is 66 and then times 1 doesn't do anything so the LCM for this one is going to be 66 all three of these numbers will fit into the 66 that's the idea behind uh LCM uh LCM questions with more than two uh numbers okay so for question o where we have four numbers uh again we have to look for all four numbers what can all four of them be divided by here two two so one four six okay at this stage uh obviously because the one right here uh there are no numbers that all four of them can be divided by so now we ask are there three numbers that can be divided by the same thing here uh four 4 six and 9 are the only ones left right we don't care about the one anymore can I divide 4 six and N by the same number no yeah it doesn't work these two are even but that's not these two can be divided by three but this one cannot so three we're done we don't have to divide three numbers by the same thing uh you'll probably do that in P yeah so if there are not a set of three numbers that we can divide by then we look for sets of two numbers that we can divide by so we skip the step where we're looking for all three are there two numbers that can yes right pick one or pick two numbers here yeah six and N both of these can be divided by three so we're going to do that the one and four will stay there because neither of them can be divided by three 6 it is 2 9 is going be three and then this is our next step right are there again we're we're on two numbers here uh four and two work cuz they're both even so I would divide by two next so the one and the three are going to stay there because they cannot be divided by two then four two is two and two is one right can I divide two and three by the same thing no because they're full time right so at this stage we finally stop okay so again look for all the numbers being divisible by stuff once that's done look for three of them and then once that's done uh in this case we didn't have any right then you jump to two and then you do that and you keep going until you stop and then now we do the whole L thing and we multiply all these numbers together that will give us our LCM so 2 * 3 is 6 * 2 is 12 and then over here on the bottom I've got 2 * 3 which is 6 I don't care about the ones so 12 * 6 will give me 72 so 72 is the smallest number that all of the these can multiply into I can take sem2 and divide by all of those and it will work so that's how you would do it with four numbers and again in question p as I said earlier uh there is nothing that will divide into all four of these so you would start by looking at what can you divide three of those numbers by and then two and then so on okay so for number 11 and I'm doing the fourth column here product is 72 I don't know the factors sum is 22 what does this mean so product again is when you yeah multiply numbers together sum is when you add them together yeah very good this is basically saying Hey I want two numbers that will multiply together to get 72 but when I add them I get 22 that's what this is saying and we got to figure out what those numbers are how do we figure out those numbers okay so there's a couple ways to do this um I did it using the listing the factors method when I did with my other block there uh this one I'll do uh like factoring like this if we're breaking um so I would typically start with one because I know everything works with one 72 divide can be divided by one I would need a 72 here right so 1 and 72 will multiply together to give me 72 so oh that's my answer right Mr Wong no because 1 + 72 doesn't give me the 22 that I need here so that is not the correct answer so I keep going what else can 72 be divided by and I would count up from this one here I would go to the two next I know that works 72 by 2 is 36 2 and 36 oh that doesn't add up to 22 okay that's not correct either and I keep counting up and I skip anything that doesn't work right but I keep counting up if the things do work so three does that work yes it does works with 24 does that work for adding up to 22 no it's adds up to 27 so that's not the right one I don't want that one four does this work with 72 yes I would need 72 4 I would need 18 hey these ones add up to 22 I know that my factors are 4 and 18 okay now let's say this number wasn't 22 let's say it was something else right then 48 would not be correct either you would keep going does five work though no so I would skip five what about six does six work yes 12 right does seven work no does eight work yes eight and nine and by here I've got nine there already so I don't need to test nine because that'll just give me eight again right so here you get a u shape that covers all the numbers that 72 can be fact and then you just look for the right one that you need to get the the sum which in this case is the four and the 18 so that's how I would approach those questions there so question number 11 fill in the missing numbers the product so when I multiply two numbers together I get 75 and the factors are the two numbers which I don't know I have to figure that out when sum so when I add the two numbers I get 20 so so what I would do is I would start with the product number the 75 there and I would break it into okay what two numbers will multiply to give me this product 75 we have to multiply with one to get the 75 right but when I add them do I get 20 no so I have to keep going so I start with this one and the actual number and then I move up uh until I figure out what the the two vales are can I divide 75 by two no so can I divide by three yes 3 * 25 will give me the 75 right like you can check this by just going 75 and then dividing by and then numbers so that'll give me 25 but that's also not what I want because three and 25 when I add them does not give me 20 right so I keep going 75 can I divide by four punch your calculator oh decimal no okay 75 can I divide by 5 yes yes five and 15 when I add these I get the 20 so the factors here will be 5 and 15 and then that's how you know that it's the right now let's say it wasn't 5 and 15 you could keep going to right you would test the six to seven to eight 9 so so on and so forth until you get to 15 because 15 is already here right then you know you've covered all the numbers uh Tom Di and Harry get half days off with pay uh Tom gets a half day off with pay every 8 days Works uh dick gets every 10 days and Harry gets every 12 days so their day off comes at different time after they've worked different periods of time so if that's the case uh they are looking for the next day when all three of them will have a day off again and then they can hang on together if all three are off together on April first oh so we got to keep track of how many days as well and get like an exact date uh what is the next date so so is this a GCF question or an LCM question LCM yeah very good because obviously the GCF we're going to get a number smaller than 8 10 and 12 and that's not going to work because the smallest sorry the GCF for all of these looks like it's uh two right but two days after no they're still working right so that doesn't make any sense so yeah this is going to be LCM after some many days from now they will all their days off will line up again so that's going to be a number that's bigger than our original numbers right the eight the 12 and the 10 so that's going to be LCM the LCM will always be bigger than the starting numbers that you have the G the GCF will always be smaller than the starting numbers that you have so let's do the uh LCM then let's do the whole setup for the L division we're going to go like this what can I divide all of these by two good so it'll be four five and six can I divide all three of these by anything no so now I reduce because again LCM you drop down to only two numbers right two numbers here that can be divided by the same thing which ones are they four and six yeah we can divide both of those by two so the four and the six will get reduced what happens to the five yeah it stays there right because it cannot be divided by the the number that we just divided by so the five is still there 25 and three can I divide all sorry can I divide two of them by the same number no yeah cuz all three of them are Prim numbers right our Prim number list so now we get finally Stu our LCM is going to be these numbers all multiply together okay so this 2 * 2 will give me 4 this is 10 * 3 30 so it looks like in 120 days right 4 * 30 120 days later uh all their uh days off will line up and then they can all hang out together now I don't know what date 120 days after April first is because someone said 31 days and stuff I'll let you figure that out the important part here about this question was the LCM and getting the 120 days