Divisibility Rules

ding-dong Michael here I hope that you are all practicing responsible social distancing right now it is so important that we all do our part in times like these we need to be unified but today we're going to divide if you want to know if a certain number can be divided by another number with no remainder left over well you could use a calculator or you could ask Google but where's the fun in that look today we are going to take a stroll through some divisibility rules for the numbers 1 through 9 these procedures are a lot of fun they allow us to quite quickly determine whether a given number can be evenly divided by another number that is with no remainder left over now a calculator is sometimes faster than these procedures I'll admit that but these procedures allow us to do two things they allow us to develop our number sense and they also allow us in these lonely times to make a bunch of new friends math friends let's get started now it's pretty easy to check if a number is divisible by one two or five because if the number is an integer if it has a no decimal or fractional component well then it can be divided into groups of one if it is even it's divisible by two and if it ends in a zero or five its well it's divisible by five now a lot of us also know a trick for checking divisibility by 3 the trick is to take all the digits in the given number and just sum them up if their sum is divisible by 3 so is the actual original number let's take a look at this in action first let's pick a number completely at random like oh I don't know how about 360 2880 is this number divisible by 3 can I take this many things group them all into threes and have nothing left over well let's apply our trick we just simply sum up the digits in the number we have 3 plus 6 which is 9 plus 2 which is 11 plus 8 which is 19 plus 8 which is 27 plus 0 which keeps us at 27 now because 27 is divisible by three it happens to be three times nine so is this original number so that helps us take care of divisibility by three what about divisibility by nine well as it turns out this trick works there too if the sum of every digit in the spelling of the number sums up to a number that is divisible by nine well then the original number is as well but what about six well for six all you need to do is make sure that it is evenly divisible by three and that it's even so in this case we know that three hundred sixty-two thousand eight hundred and eighty is divisible by 3 because this trick shows us that but it is also even therefore it is divisible by six as well okay now I think it is time for us to move on to divisibility by four we'll keep using the same number three hundred and sixty two thousand eight hundred and eighty can I divide this into groups of four with no remainder well here's how you can check if the last two digits of a number form a number that is divisible by four then the entire number is eighty is easily divisible by four right that's just 4 times 20 now let's say though that we had a different number like three hundred and sixty-two thousand eight hundred and ninety six let's say I don't know right away whether 96 is divisible by four well here's a little bit of a trick you can use to figure out if it is simply take the tens digit of this two-digit number and double it which means two times nine and then add to that the ones digit six if their sum is divisible by four then the original number is as well in this case we have two times nine which is 18 plus six this gives us 24 and 24 is divisible by four so both our original number and this new one can be divided into groups of four with no remainder pretty cool what about eight well if I want to know if a number can be divided evenly by eight I need to look not at the last two digits but at the last three here's what I do I take the hundreds digit and I'm old apply it by four so this is four times eight to that I add the tens digit times to the tens digit is eight and then to that I add the ones digit zero 4 times 8 is 32 2 times 8 is 16 and 0 is just 0 now 32 plus 16 is 48 is 48 divisible by 8 yes it is it happens to be equal to 8 times 6 so our original number is evenly divisible by 8 okay so we have covered divisibility by 1 2 3 we've done 4 & 5 & 6 & 8 & 9 whoa-ho-ho what about 7 well 7 is very very fun here's what you do with 7 I think that for a number as long as our starting number which has 6 digits we're going to probably have to do this procedure multiple times but all of these tricks can just be done repeatedly you just do it once to the original number and then if you still don't quite recognize what you're left with and whether it's divisible by the target number or not guess what just keep applying the trick until you see something you recognize we're gonna definitely have to do that when we check divisibility by 7 for 350 2880 but here's all you have to do take a look at the number and look at the ones digit now take that ones digit and multiply it by 5 okay and then add to this what's left as if it's a number all by itself which means we're going to add 5 times 0 2 3 6 2 8 8 all right now we sum these up and if their sum is divisible by 7 then the original number is as well so what do we got here well we've got this number plus 5 times zero which is zero so it's just the same number now I'll be honest I'll admit this I'm not afraid to I don't know if this is divisible by 7 or not actually I do because it's not a coincidence that literally every single one of these digits evenly goes into this number 360 2880 happens to be nine factorial which means it is equal to one times two times three times four times five times six times seven times eight times nine therefore it is a multiple of all of them but let's show that using this trick I can apply the trick again to the result here all that means I need to do is I take the ones digit I multiply it by five so that's five times eight and then I add to this expression what's left which is just three six to eight now five times eight is equal to 40 and this number is three thousand six hundred and twenty-eight if I sum them together what I get is three thousand six hundred and sixty-eight alright if I still don't know if this is divisible by seven I just keep going so we take the ones digit alright and we multiply it by five eight times five again as we know is 40 and then I add what's left three hundred and sixty six okay this gives us four hundred and six is four hundred and six divisible by seven if I still don't know we just keep going what is six times five well we'll deal with that later we just know that we need to take five times the ones digit and summit with what's left 40 now 5 times 6 is 30 and 30 plus 40 is 70 70 II's aliy recognized as being divisible by seven it is just seven times ten therefore because this is divisible by seven our original number is two pretty cool very fun but how do these tricks work oh well that's the fun part let's get into it let's start with divisibility by three why should it be the case that summing all of the digits used in the spelling of a number and then checking if that sum is divisible by three tells us anything about the divisibility by three of the original number well let's generalize this and just imagine a number that has let's say three digits this works for you know a number with any number of digits though but let's say that this is our three digit number ABC where a is the hundreds digit B is the tens digit and C is the unit's digit now what ABC means when we write it down we don't actually mean yet ABC that means you know a number of things equal to a plus B plus C no no instead we use a base-10 positional system where this actually means 100 times a plus 10 times B plus C so if this number is say 417 well then we're going to have four 100's 110 and seven units but let me ask you this what would it take for this the sum of the digits in the number to equal the actual amount the number represents well we'd have to add more stuff to this we only have one a here but we need 100 so we're going to need 99 Moraes now we only have one B but we need 10 so that means we're gonna need nine more B's we already have a seat so we don't need to add any more C's there you go this right here is equal to this the actual amount represented by the original number oh look at this this is very interesting 99 a plus 9b this part of the expression will always be evenly divisible by three no matter how many digits are in the number that you are checking every coefficient in this part will just be a string of nines which means each term in it will be divisible by three and if you add a bunch of groups of three to another bunch of groups of three well what you're left with is just a bunch of groups of three what you're left with is a sum that is still divisible by three and so because this component is always divisible by three all we need to do is check whether this part is if it is well then the original number is the sum of two numbers divisible by three so it itself will be visible by three in a similar way the divisibility check for four operates as well so let's check that let's again say that we have a number like a and B and C now remember that when we check for divisibility by four we only concerned ourselves with the last two digits why would that be well it is because conveniently four goes into 100 an integer number of times 25 in fact so any number that ends with at least two zeros is divisible by four for example the number let's say one seven nine two zero zero I just made this one up but I know that four will evenly go into this because it is just 1792 hundreds and four goes into a hundred what that means is that if we when we take a number like the number that we started with 360 2880 it doesn't end with two zeros but we can think of it as being three hundred and sixty-two thousand eight hundred which we know that 4 goes into plus 80 so we only need to check the divisibility of this component those last two digits 80 all right so remember we took BC right we took the the last two digits and we doubled the tens digit and then we added the ones digit why does this work well in the same way that the check for divisibility by three works let's remember that BC we're forgetting about a for now BC means 10 times b plus c number of things well how do we get from this checking expression to the actual value of BC well we're going to need eight more B's and no C's we'll look at that 8 8 b 8b will always be divisible by 4 because 8 can be put into groups of 4 and we're gonna have b times as many of those groups of four so we only need to know if this other part is divisible by 4 and if it is well then the end higher number is you might think oh why don't we just use this trick to check for divisibility by eight I wish that we could but the problem is that this entire trick relies on the fact that the last two digits of the number are all that matter because four goes into 100 but eight does not however eight does go into 1000 eight times 125 equals 1000 so if a number ends with three zeroes or more we know that it is evenly divisible by 8 we'll need to worry then about the last three digits of any number to check for divisibility by eight if you're thinking about a number that doesn't have three digits that it's only a two-digit number well then you know you you probably can just figure out if it's divisible by 8 by simply knowing your 8 times tables if you know them if you've memorized them up to eight times ten at least well that's already 80 from there you go from 80 to 88 and then you go to 96 and then you're done those are the only other two-digit multiples of eight 96 is eight times 12 of course but when it comes to a three-digit number I'll say ABC doesn't matter what other digits are out here if any we check divisibility by eight by taking four times the 100's digit and summing that with two times the tens digit and then just the unit's digit by itself why does this allow us to check for divisibility by eight well let's take a look we know that the actual amount represented by a BC is equal to 100 times a plus ten times B plus C to get from this expression the checking expression to the actual amount represented by ABC we need 96 mores we need eight more B's and no more C's what we just talked about how 96 is 8 times 12 and of course 8 goes into 8 so this part is always divisible by 8 regardless of what a and B are so we only need to make sure that this expression is divisible by 8 if it is then the original number is okay you know what all we have left to discuss now is divisibility by seven why did that trick work it was kind of strange right we took the ones digit of the number multiplied it by five and then added it to what was left well the trick to kind of wrapping your head around why this works is to figure out what it means to sum what's left after removing the ones digit so let's say that we have a number like a b c doesn't have to be three digits long could do it of any length but we take this ones digit c and we multiply it by five and we add that to what's left how do we represent this procedure of taking what's left and treating it like it's its own number well one way to do that is to think about the original number a b c and then subtract C from it okay this would allow us to take a number like 417 and then subtract that last digit seven if we subtract seven then we're left with 410 we've turned the ones digit into a zero then we we want to just get rid of it completely and we can easily do that by dividing by ten in which case we then wind up with 41 so we're going to have to divide this by 10 which is the same as multiplying by 1/10 this looks a little bit messy and I hope you're able to follow it all we're trying to do is algebraically show what is done when we ignore the ones digit and treat what's left is its own number that's what we're doing here we're taking away the ones digit so it becomes a zero and then dividing by 10 so we turn ABC into just a be perfect now how does this and whether it's divisible by seven tell us that this is also the original number is also divisible by seven oh boy well this is very fun let's play around with this expression and see what else it is either equal to or divisible by the same numbers as okay so let's distribute the 1/10 first because 1/10 times the stuff in parentheses can also be represented as tenth time's ABC minus 1/10 times C all right then we still have to add the five see at the end oh look at this we've got ourselves two different amounts of C we can combine those what is five C minus 1/10 of a C well that's five minus 0.15 minus point one is a positive four point nine we still have this 1/10 ABC so this expression right here this third one that we've got is equal to this expression which is equal to this expression that we actually are using to check for divisibility by seven you might already have a bit of a sense of why this works because 4.9 looks an awful lot like 49 a number that is famously divisible by seven let's go ahead and make this even easier by taking this and multiplying it by ten multiplying by ten doesn't change what this can be evenly divided by it just means that whatever amount of groups we can make we're gonna have ten times more of them right it doesn't add any remainders so we're fine if we multiply this whole thing by ten that means we're multiplying 1/10 ABC by ten ten times 1/10 is one so we've got a B C and then 10 times four point nine well that's just 49 C oh look at this alright so this thing that we've wound up with is divisible by the same numbers as this which is divisible by the same numbers as this which is divisible by the same numbers as this the actual procedure we use to check for divisibility by seven but look at this guy this is pretty clearly made up of a part that is always going to be divisible by seven 49 49 things can be put into groups of seven and we're gonna have C times more of those groups so if this if this if this and if this happen to be divisible by set well then that means that ABC must be too but ABC is the original number so if this checks out and gives us a number divisible by seven then the number we started with was too beautiful I love this kind of stuff if you've got some time while you're quarantined I highly recommend playing around with numbers and coming up with some of your own tests of divisibility it's very fun I did it for six and I loved it obviously this doesn't necessarily make math faster to do a calculator will be faster almost all the time except if you've if you've memorized things but the point isn't speed the point of these exercises is to develop our number sense and come closer to this thing we call number now I'd like to end with a word from our sponsor Vsauce those of you who subscribe to the curiosity box thank you thank you so very much I hope you know how much your support helps us here at Vsauce and in my opinion helps make the world a better place it's because of your support that we are able to make things that otherwise wouldn't exist and they are things that I am very proud of things that I'm glad for the world to have the latest box I'm just thrilled with something that we were able to make and include in it in 1933 Theodore Edison son of Thomas Edison invented a puzzle that he called the caliber on 12 and we went back to the original took the measurements and created a replica of the caliber on 12 look at that whoop very beautiful the pieces are all of the same size as the original and they're even made of the original material bakelite now I'm excited about this puzzle because it is I mean totally serious it is the only puzzle that I own that I have yet to solve I have a wooden version of this puzzle that's a little bit different than the original and I've had it for a couple of years and I still have not figured it out the puzzle goes like this you take these 12 sort of burgundy pieces and you need to assemble them into a rectangle that was the original puzzle but the puzzle came in a box just like this and the pieces all came flat in the box but this box's shape is not the shape of the rectangle that the 12 burgundy pieces fit into this is a different dimension rectangle and in order to make these twelve pieces fit into here you need to use one of these three black pieces which Edison called spacers so all twelve pieces came in the original box along with a spacer so it would be something like this and somehow all 12 of these burgundy pieces and this 13th black spacer piece can be fit into the box once you've solved that puzzle replace this spacer with a different spacer like this one and now try to fit them all in it's possible when you're done with that puzzle then fit this nice little skinny nude in there and try to fit every piece into the box again that's three different puzzles but the original puzzle was just to take these twelve with no spacer and put them into a rectangle now what makes me so excited is that this is actually this actually comes with more puzzles than the original because when you bought the original back in the 30s from Theodore Edison it only came with one spacer and you didn't know which type of spacer you were going to get we have reproduced all three of the spacers he made and so now you can feast your brain on four different puzzles I'm glad to have brought this back it's a fantastic invention and as you know a partial of all proceeds from the Curiosity Box are given to Alzheimer's research so far we have been able to give more than $100,000 to Alzheimer's research that is why this box matters so much to us it is good for everyone's brain it's good for yours it's good for other people's thank you for your support we really appreciate it it helps us out here at Vsauce and I honestly believe it makes the world a better place so thank you thank you for being curious and as always thanks for watching

Channel: D!NG

Views: 1,145,306

Rating: 4.9629655 out of 5

Keywords: dong, lut, vsauce, vsauce2, vsauce3, michael stevens, kevin lieber, jake roper

Id: f6tHqOmIj1E

Length: 24min 32sec (1472 seconds)

Published: Thu Mar 19 2020

Reddit Comments

[u/wildgoosespeeder]

Michael has seen a thing or two about social distancing and quarantine from a Mind Field pilot episode.

♪Hello darkness, my old friend♪

[u/altoidbarista]

Ok!

[u/ethiczz]

Hello dingdongers