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  • 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, with 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're going to take a stroll through some day visibility rules for the numbers one through nine.

  • These procedures are a lot of fun.

  • They allow us to quite quickly determine whether a given number could be evenly divided by another number that is, with no remainder left over.

  • Now.

  • Ah, calculator is sometimes faster than these procedures.

  • All 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 12 or five, because if the number is an integer.

  • If it has no decimal or fractional component will, then it could be divided into groups of one.

  • If it is even, it's divisible by two, and if it ends in a zero or five, it's well, it's divisible by five.

  • Now a lot of us also know a trick for checking to visibility by three.

  • The trick is to take all the digits in the given number and just sum them up.

  • If they're some is divisible by three.

  • 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 362,000?

  • 880?

  • Is this number divisible by three?

  • 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 three plus six, which is nine plus two, which is 11 plus eight, which is 19 plus eight, which is 27 plus zero, 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 the visibility by three.

  • What about the visibility 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 will, 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 362,880 is divisible by three.

  • 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 Div is a bility by four will keep using the same number.

  • 362,000.

  • 880.

  • 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 80 is easily divisible by four, right?

  • That's just four times 20.

  • Now, let's say, though, that we had a different number like 362,896.

  • 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 they're some, 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 24 is divisible by four.

  • So both our original number and this new one can be divided into groups of four with no remainder.

  • 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 multiply it by four.

  • So this is four times eight two that I add the tens digit times to the tens.

  • Digit is eight.

  • And then to that, I add the ones Digit 04 times eight is 32.

  • Two times eight is 16 and zero is just zero.

  • Now 32 plus 16 is 48 is 48 divisible by eight.

  • Yes, it is.

  • It happens to be equal to eight times six.

  • So our original number is evenly divisible by eight.

  • Okay, so we have covered visibility by 123 We've done four and five and six and eight and nine.

  • Ho, ho, ho.

  • What about seven?

  • Well, seven is very, very fun.

  • Here's what you do with seven.

  • I think that for a number, as long as our starting number, which has six digits, we're going to probably have to do this procedure multiple times.

  • But all of these tricks could 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 recognise?

  • We're gonna definitely have to do that when we check the visibility by 74 362,880.

  • But here's all you have to dio take a look at the number and look at the ones digit.

  • Now take that one's digit and multiply it by five.

  • 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 five times 0236 to eight eight.

  • All right now we some these up.

  • And if there's some is divisible by seven, then the original number is as well.

  • So what we got here?

  • Well, we've got this number plus five 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 seven or not.

  • Actually, I do because it's not a coincidence that literally every single one of these digits evenly goes into this number.

  • 362,880 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 36 to 8 now, five times eight is equal to 40.

  • And this number is 3628.

  • If I sum them together, what I get is 3668.

  • All right.

  • If I still don't know if this is divisible by seven, I just keep going.

  • So we take the ones digit, all right, and we multiply it by five, eight times five again as we know it's 40.

  • And then I add what's left?

  • 366.

  • Okay, this gives us 406 is 406 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 five times six is 30 and 30 plus 40 is 70 70.

  • I easily recognize as being divisible by seven.

  • It is just seven times 10.

  • Therefore, because this is divisible by seven, our original number is too 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 the visibility 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 day visibility 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 of 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 see is the units 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, no, 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 +41 hundred's, 1 10 and seven units.

  • But let me ask you this.

  • What would it take for this?

  • The some of the digits in the number two 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 more A's, and we only have one B, but we need 10.

  • So that means we're gonna need nine more bees.

  • We already have a C.

  • So we don't need to add any more sees.

  • 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 nine b.

  • This part of the expression will always be evenly divisible by three no matter how many digits are in the number that you were 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 divisible by three.

  • In a similar way, the day visibility check for four operates as well.

  • So let's check that lets again say that we have a number like A and B and C now remember that when we checked for Div is a bility 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 atleast two zeros is divisible by four.

  • For example, the number Let's say 179 to 00 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 100.

  • What that means is that if we when we take a number like the number that we started with 362,000, 880 it doesn't end with two zeros.

  • But we can think of it as being 362,800 which we know that four goes into plus 80.

  • So we only need to check the day visibility of this component.

  • Those last two digits 80.

  • All right, So remember we took BC Right.

  • We took 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 Div is a bility 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 B C?

  • Well, we're going to need eight more bees and no sees.

  • Well, look at that.

  • Eight eight be.

  • Abie will always be divisible by four because eight can be put into groups of four and we're gonna have to be times as many of those groups of four.

  • So we only need to know if this other part is divisible by four.

  • And if it is well, then the entire number is you might think, Oh, well, why don't we just use this trick to check for the visibility 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 zeros, orm or we know that it is evenly divisible by eight willing to worry then about the last three digits of any number to check for the visibility by eight.

  • If you're thinking about a number that doesn't have three digits, that's only a two digit number.

  • Well, then, you know you you probably can just figure out if it's divisible by eight by simply knowing your eight times tables if you know them.

  • If you've memorized them up to eight times 10 at least well, that's already 80.

  • From there you go from 80 to 88.

  • When you go to 96 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 air out here, if any.

  • We check the visibility by eight by taking four times the one hundred'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 two visibility by eight?

  • Well, let's take a look.

  • We know that the actual amount represented by A B C is equal to 100 times a plus 10 times B plus C.

  • To get from this expression, the checking expression to the actual amount represented by ABC.

  • We need 96 more A's.

  • We need eight more bees and no more.

  • Sees what we just talked about, how 96 is eight times 12 and of course, eight goes into eight.

  • So this part is always divisible by eight, regardless of what A and B are.

  • So we only need to make sure that this expression is divisible by eight.

  • If it is, then the original number is okay.

  • You know what?

  • All we have left to discuss now is divisible ity 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 toe what was left?

  • Well, the tricked it kind of wrapping your head around.

  • Why this works is to figure out what it means to some 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 to do it of any length.

  • Um, but we take this one's digit, see, and we multiply it by five, and we add that tow 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, ABC and then subtract.

  • See 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 10 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 algebraic Lee 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 the 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 numbers 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 1st Because 1/10 Time's The stuff in parentheses can also be represented as 1/10 times ABC minus 1/10 times c All right, then we still have to add the five c of the end.

  • Look, this we've got ourselves two different amounts of sea.

  • We can combine those What is five C minus 1/10 of a C?

  • Well, that's five minus 50.15 minus 0.1 is a positive 4.9.

  • We still have this 1/10 ABC.