Divisibility Tricks - Numberphile - VoiceTube: Learn English through videos!

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Obviously, I put along my number file playing cards or they're playing.
Cards are available for these are better on this.
Okay, so just Well, you know, we know what number five playing cards.
There's me.
Of course, I always itself.
In fact, Brady there is there, is you?
Of course they can watch.
They're just here to watch.
I'm also gonna let Matt watch because Mark is Thea, is this king of number five card tricks, right?
So Okay, so what?
I've just taken nine cards, Okay?
The first, all the clubs, basically all the numbers clubs.
Here's the challenge to you, Brady.
I want you to put them in any order you like.
I'm not even going to pay attention.
Let's go.
That will do, mate.
Yeah.
Okay, so we read off.
What number that's gonna bay 945,136,872.
Do you know anything about that?
Numbers?
Anything.
You you know, it's about it.
It's even.
It is even.
Yeah.
You think it's something not prime.
Then do you think these numbers in the nine times table probably not.
Don't feel like it is?
Yeah.
I've been wanting nine numbers is in the nine times table.
Right?
So it's you might think it's unlikely.
Sure.
We should.
We have a bet.
Okay.
I'll bet you're fired, right?
You're saying it's not?
I'm gonna say is Okay, so should we check?
Okay, let's stick.
Right.
So Okay, so we put the number in divided by nine, see if we got a whole number.
Oh, yes, we do.
Okay.
Do you want to go again?
Don't try again.
You can even take it.
You can take a number out of you.
Any number you take on him, right?
All right, let's take out.
Double.
Quick.
Straight together.
Top number.
Okay.
You think out something, mister, You've been left with this number, okay.
To recognition.
You can write.
What, you want to move them around a bit.
Yeah.
Let's swap the five on the fourth day with you and show you.
You done?
Yeah.
Okay.
You think this is in the nine times table?
Yes, I did.
You did idea.
Okay, well, you might have won your battle.
Okay, I study.
Check it.
Divided by Let's see nine equals.
Yes, it is.
Well done.
Well, you removed the nine, which made it a lot easier for me if you'd removed.
Say, let's let's put this back.
If you removed the five, my next step is gonna be so It's only fair I get to move on out, take one out as well.
And I was gonna take out before we'll come back to.
Why, that that work later.
Try another trick.
Okay.
Slightly different one.
They've got the 1st 4 cards now wants pulls.
The three are quite common, right?
I mean, one in three numbers is a multiple of three after new four cards.
All right, you can put him in any older just just make me a multiple of three.
Well, you just try it, I guess.
All right, let's get with it.
21 for three.
Okay.
You don't sound confident.
Radio like, save it confidence like a 2 to 143 when divided by three.
Oh, dear.
You need some help?
I can get health.
You?
Yeah.
I'll just I'll just gonna put one card and card in for you, right?
This one.
I know you can put it whatever you like.
Let's put it in the middle.
Okay, let's put it in the middle right.
You wanna put right in the middle?
You sure you don't put over here now?
I'm over in the middle.
Okay.
All right.
Well, air, let's turn this car over.
Okay?
Now, let's see if this helps the situation.
Okay, let's try it now to 1543 has tried Divide in that way.
Three.
It worked.
Okay, so what is going on Radio?
What is going on?
Well, these little tricks Well, I kind of inspired by actually my own incompetence.
And because in our last video that you and I did together, which was about Belle figures Prime.
You remember we asked the following question we asked, Is this number?
We had a one and then lots of zeroes.
And then a 616 And then another bunch of zeroes.
And then one.
We asked if this number was a prime number or not.
We said it, you know, reporting enough zeros.
Can we Can we find a prime number on?
I ran a little computer code and got up to 100,000 zeros and didn't find any of them.
If you remember this, I saved up to anything.
100,000 and no prints.
And, of course, our super smart number.
File viewers immediately saw that this was never gonna be a prime number.
And the reason?
Waas Well, if you calculate the sum of all the digits, okay, so the cross, some of this object, then you'll get one plus lots of zeros.
Six plus one plus six butcher zeros again, plus one on that Totals 15 now 15 is divisible by three.
And if that number, if the cross some is divisible by three, then it guarantees that the original number is divisible by three.
So you were you were running this code checking every number when you would never It was never gonna be There's never gonna be They're wasting waste of confusing time.
Break time.
I should have just been smart number five years, right?
You think I just I just, uh to be honest, if I'm honest, I was trying to load the different things because I wanted something that might sort of cool in the video.
And I got so sort of, you know, sort of hung up on on all the different things I was trying.
I sort of lost sight of, of the beauty of maths Because this explains what went wrong with the with the with your your attempt of the att The number earlier, right?
Okay.
You took the 1st 4 numbers.
Whatever.
No combination you take these in.
I mean, these numbers 123 and four They add up to 10.
So whatever order I put them in the cross song is always gonna be 10.
But 10 is not divisible by three.
So whatever number I create cannot be divisible by three.
And now I at the five and it doesn't matter where I put it.
The cross Some is now guaranteed to be 15.
Whatever number I create is guaranteed to be divisible by three.
Okay, If the cross so Mr Visible by three the number is divisible by three.
It's also true for nine's.
If the cross, um is the visible by nine.
The number is divisible by night on that comes back to this one first nine numbers What?
Today?
Something they sort of 45 45 is the visible by nine.
So wherever combination of these numbers, I take whatever order a take guaranteed to be divisible by known.
This isn't true for other cross sums like if the cross some is divisible by four.
No, it doesn't mean the number.
I don't know.
It's just 33 and nine.
It works for Okay, It works for three and nine.
Okay, there are this or the rules that you can apply to other numbers, and we'll get into that.
But for this, this cross, some rule, it works for threes and nines.
Let's see if the roof Okay, let's see the proof.
Let's write down the kind of number that we might be interested in.
Let's just write it down as follows.
So the number and that's just right.
It's and it's gonna be an M plus one digit number.
Okay, Which we might write like this.
Okay, so we got a zero in the ones they won in the tens and so on.
Let's actually write down what this number is sort of algebraic.
Lee.
So what?
This means that end is really tend to the end times A and plus tens of the n minus one times and minus one plus 10.
A one plus a zero.
Okay, so that's what this No Berries.
Okay, Any number in principle at this stage now, Let's write down the cross.
Um, okay, so the cross comes easy to write down.
Let's call it tear, then.
Okay.
That's just gonna be the sum of all these individual positions and plus and minus one plus two.
A little toe, a one plus a zero.
Okay, Now let's do that.
Take away that.
Do that.
We can end minus tear, then, Okay, let's see what we're going to get.
So in this position will get tensed.
The end minus one and plus 10 to the N minus one, minus one and minus one.
You'll get similar things all along here.
You'll get 10 minus one, a one and then the a series of counsel.
Now look what you've got.
You've got This is nine, right?
The next one in the sales will be 99.
This one is going to be a whole bunch of nines.
This one's gonna be a one more nine.
You'll have another nine there, So there's gonna be all the nines, all the nines.
There are gonna be nines.
So they're all divisible by nine.
Okay.
So I can pull out a factor of nine.
So this is always gonna be nine times some whole number.
If the cross, um, is divisible by nine, this is the visible by nine in this better be divisible by.
And similarly, if the cross, um, is divisible by three this is definitely divisible by three.
Then this number, the original number must be divisible by three.
That's the proof.
That simple.
Right, then easy to prove.
Okay.
Why does that any work with nines and threes?
Precisely because the fact that you take the factor that you take out is nine.
Okay, that's why So, you know, basically, you need it for threes and nines.
So they're a bunch of these rules, some, some of which are easier than others.
And we can go through them, right?
And so one.
Okay, that's he's pretty.
He's the second.
Something's divisible by one tear is easy.
It just has to be an even number three's.
We've done about fours.
OK, what's the visibility rule for four.
Okay, I'm gonna write you down a multiple of 4145632 viewers can check of.
That's a multiple of four directly.
We know it is.
How do we know it is the way you do the full check is you just look at the last two digits, okay?
And you ask yourself, Are the last two digits divisible by four?
If they are, that's enough.
So here it's 32.
That's divisible by four.
I don't need to worry about the rest, and it's easy to see why.
Because these air hundreds thousands tens, thousands 100 is divisible by 4000 divisible by four.
And so on all the all the higher powers of 10 of the visible by force and the only thing you need to check of the last two.
So four they're easy as well.
It's a really easy test.
Five Pretty easy things.
Ending 5006 is well, there you have Thio do the two tests on the three test.
Okay, so you check is even in any deal.
You across some test for threes and it has to pass both.
It has to pass both Yes, seven's.
We're gonna come back.
The 777 is a notoriously magical and mystical number.
Eight and seven are the hardest of these air.
You know, the sort of lower numbers for doing the visibility test will come back.
The seven layer eights Let's try eights.
Okay, so here's an example.
1752464 So again views conjecture at this.
This is indeed a multiple of eight.
Directly.
How do we check in?
Well, it's very similar to the four test, but it comes in an extra stage.
But depends how good your mental arithmetic is, actually.
Okay, so what you do is so where is with fours?
We check the last two digits with eight.
We check the last three digits.
Okay, so we look at this 464 and we ask, Is that guy divisible by eight?
If that guy's divisible by eight, then this guy's guaranteed to be divisible by eight again.
Why does it works Very similar reason to the to the force.
So basically, because the thousands, the 10 thousands and so on, all the higher powers attempt are automatically divisible by eight.
So you don't need to look at the last three, but then about you, Brady.
But my mental reference it's not good enough to check of straightaway whether that number is divisible by eight.
But there is a test for check in very easily.
But if you've got a three digit number, whether it's divisible by eight.
Let's take this case for 64 What we do is we take these two digits here, okay?
46 and we multiply that by two.
And then we had the other digit.
Okay, if you do that, you get 96.
Now, my men within sick is just about good enough to know that that is a multiple of eight.
Okay, so if this number is a multiple of eight, then this number is a multiple of eight on by association servants of this number.
So it's a two stage test.
So let's just just prove this three digit test for the multiples of eight is again quite easy.
So we just take and on the number that we're interested in, Let's write it is 100 times 82 plus 10 times a one plus zero.
The test we're gonna perform is called it.
Tien is Basically we take the 1st 2 digits, so that's like 10 A two plus a one.
Okay.
And we multiply it out by two.
And then we had the other one, which is easy room.
Now take these two away from one another, and you get end minus Tien.
What are you gonna get your getting?
100 minus 28 two's.
That's 80 a two.
You get 10 minus two A ones that eight A one and a zero is canceled.
You can probably see it now, right?
Basically, if this number is divisible by eight, which is the thing, that which is the thing that was our test.
So if the national mood is divisible by eight, this is guaranteed to be divisible by eight.
Then that's gonna be divisible.
Wait, that's why it works.
Shall we carry on?
OK, you know what?
We don't nines already.
So tens.
I think you know, most of you can work out tens living as we do in our in our base 10 world.
So let's do the test for 11 dead.
Simple, really easy test.
Okay, so let's pick a number.
This you can check.
Even usual.
Calculated.
This is a multiple of 11.
Okay, so let's do the test.
Okay?
So the first thing you do when you do the elevens test that you have to reverse the number.
Okay.
So he reversed the order of the number, so you write.
It was 9665205 And now you don't take the cross.
Some of this number you take the alternating cross.
What I mean by that?
Well, that means I do.
Nine minus six plus six minus five plus to minus zero plus five.
Then I think it is summer.
Look, think it's 11.
Okay, so, again, the rule is if the number you end up with here is a multiple of 11 then the number you started out with is the world's full of 11.
So if it had been 22 way got 22 year, that would have, Then we would have guaranteed the original novel was a multiple of 11.
Gosh, that's really obscure.
Yeah, we could prove it if you want.
It's quite easy to free, but there's another sort of flight.
So take on the card trick that you can do for this one.
It needs you to be slightly quickly the with your mental arithmetic.
So basically, you lay out a bunch of cards.
I don't know any order.
Let's Let's not worry too much about what it is, okay?
And let's take 55 digit number.
Okay, So you want a staff with something that's not gonna be a multiple of 11 careers.
Check out of this one is or isn't it?
Before we proceed with such a little near its top Good.
Just laid out random number here.
Okay?
And now you're gonna show our clever you are, But you're just gonna say I'm gonna add one card and make it a multiple of 11.
Okay, So how do you do that?
I can't do that thing in my head reversing and alternating Cross some.
Okay, let's see if I can do it.
Okay.
Said Well, okay.
This is a challenge against it.
That is what you have to do.
You have to do it.
Yes.
You have to leave the attitude in your head.
Okay, so I'm gonna Okay, just just take it.
Worst thing isn't rubbish.
Okay, Right.
Yes, I know what he thought.
Okay.
Yeah.
Okay.
So I need to, uh okay.
I think that should do it.
Five stars.
Yeah.
So you got much for your mental arithmetic is bad in mind.
You'll do it in no time.
So you were reversing that number, then?
Alternate?
Yes.
I was adding I was doing the cross.
Um at the alternating.
Cross them in reverse.
Right, five minus three plus one, minus two plus four, I think is five.
Okay, so I need to take off.
That would get me to zero, which is definitely a multiple of 11.
Alright, your goal is to get some water, pull a villain.
You can also start playing around putting.
Norm is at the start, but it gets a bit more complicated, but all right, strike.
Okay, five for two.
135 divided by 11.
Okay.
Okay.
Obviously, you need to be pretty sharp.
The mental arithmetic to do this one, which I'm afraid I'm not 12 twelves.
Well, twelve's just you build it from the fours and threes.
OK, so it's the same as he just You do the four test and then you do the three tests.
Oh, that's easy.
And that just has to pass.
Both asked apart supposed here.
We could go on and let's come back to seventh cakes.
We skip sevens, which is a bit naughty.
Sevens are hard 17 notoriously hard.
There is a reasonably simple one.
And it also reminds me a friend of mine I got Frank Aled.
Nori.
Nori is a lot of things, but he is not a mathematician, but he has.
This is this thing you can do where if he sees a number, he can tell immediately whether it's Ah, multiple of seven on.
It's really strangely sort of like sort of seasonal bring a cloud in his mind.
And then if you sort of warm towards if he feels like a friendship towards it, then that means it's a multiple of seven.
This is totally Riel Distillery.
He's not.
As I said, he's not mathematician.
He did English uni.
Yeah, it is.
He has this rather unusual thing that he has.
How did you find out that he has this ability?
How did it come up in conversation?
Nori likes to talk about himself a lot.
It was always gonna call.
If it's some point, do you test him?
Do you get it?
I think he can do it.
There's no listen about Totally.
Oh, yeah, So you can go to about five digits and then he starts to feel physical effects.
Apparently, at that point, you should be making video with Norrie.
Oh God, you really are an iwas already was what I told him I thought I mentioned this.
He decided that the whole video should be about here.
This is typical Nori that I said, hold.
It is not gonna be about, you know, Rory.
But maybe it is.
Hey, let us have a picture of him to put on the screen, and I will love to have a picture of himself on the screen.
Hey, so how'd OH, idea.
Okay, so we got a multiple of seven.
Let me write it down.
Norrie looked at that number, he'd start to seal his friendship, feelings of warmth and all that.
Right.
Okay, So the seventh test, the simplest one I could find is the following you take the blocks of three.
Okay, so this block of three here, 984 and then we do like an alternating some again.
So we know.
Take the next block of three.
Which is this one.
We take it away.
976 And then we add, we keep going like this.
The next block is gonna be a six.
Of course, we stop here, Okay?
If we had more, we keep doing this in in an alternating way.
Okay, So now you actually calculate this thing, which is obviously pretty easy to do.
So that is gonna be, I think, 14 if the result is a multiple of seven, that guarantees that the original number is a multiple of seven.
I imagine quite often we're going to get negative numbers on this through this test, that sign that they can be negative numbers.
It could be a multiple of seven.
It's just that my negative number, times something this was actually a really, really simple example.
I was pure Look, that it turned out to be so simple.
Normally you're gonna end up with a three digit number here, right?
So you're faced with the question of is that three digit number a multiple of seven, which generally is not gonna be that easy.
And so let's do another example, maybe quickly, just to see how that can arise.
Here I have another number which happens to be a multiple of seven.
So I do 123 minus 872 and I'm gonna get this attends out to be minus 749 That's the art.
Isn't it really hot beyond my mental arithmetic, that's for sure.
Okay, so good.
Now do another 33 digit test on seven.
Is that really?
But I mean, 749 actually is pretty easy.
Really easy thing.
This is why I missed the belphegor, private.
Good.
Let's pretend that.
Okay, Okay, but let's pretend it's hard.
But what will be the test that we do?
Okay, so you take your seven.
We don't need to worry about you.
Take a 749 Okay, we take this number here, which is 74th on, we take off twice this number here.
Okay?
Said two times nine.
Okay, that's 56.
And then the issue is is this a multiple of some?
Which, of course it is.
Okay, that would guarantee that this is by association.
This is this This test for three did.
It is quite straight forward to prove proving that these blocks of three works is not so easy.
You can prove it.
Okay.
I hope you can prove you should be using.
You can't prove it.
And it relies on two facts when I'm gonna do the proof.
The first fact that you need to prove this is that this number is basically seven times 11 times their team.
A case, actually What's important about it is that it's a multiple of seven.
Thea.
The fact that you need to know is that 999,000 999 is also a multiple of certain, and it's this multiple of seven.
So this is a multiple of seven.
This is a multiple of seven.
These are the two facts you need to prove this set particular seven's test, but it's quite an elaborate proof it yet it z yeah, exactly.
It's it's It's not It's not particularly entertaining proof notice, by the way, it's got this number in here.
Remember that?
I do.
Isn't that the one where you can like cycle?
So yeah, yeah, we did way did a video of the fight.
I think we even probably showed that result.
I'm quite pleased me that that reappeared.
Okay, so there is You can do one in old generality.
Okay?
And which I don't know how exciting this is, but we can do it with an example.
Way to test any number that ends in 137 or nine.
You might need to use a few alterations off it to actually get get down.
T the bottom but yeah, there is a general test, so let's let's do an example.
Okay?
So I can illustrate that the general test on Yeah, I know.
So let's try it for 23.
Okay?
So the first thing we have to do, So let's start with 23.
That's the number of visibility.
Yeah.
This we're gonna look, we want ask things in the 23 times table we know we're gonna test for is this number, which is in the 23 times table.
So the test goes in various stages.
First thing you gotta do you gotta generate a new number from this guy.
Okay, I'm gonna call it Davina.
Okay.
From 23 from 20.
300 generate 23 different number.
The first thing have to do you have to multiply this by something to make it end in nine.
OK, so if you've got a number ending in three to make it ended, and nine, you just multiply it by three.
Okay, so that will give us 69.
So you gonna have got number that ends in a nine.
Okay.
If you had your number ended in a one or a seven or a nine.
So it wasn't 23.
It was 21 or 27 or 29.
You had multiplied by a different name.
It to get it to end on nine.
Okay, Would still be one of 137 and nine.
But you just do something else.
The goal is to get some of the end of the night.
Okay?
Once you've got someone ends in a nine.
You then add one.
Okay, so that's easy.
That gives us something that's a multiple of 10.
It's guaranteed to be a multiple of 10.
Okay, In our case, it's 70.
Now, once you've got this multiple of 10 you just divide by 10.
Okay, so let's divide this by 10 and we obviously get seven.
Okay?
This is our davey number.
So what's the test?
Okay, so now we wait, we've got Davey number.
We've got our number that we're interested in looking at and say The thing you have to do is you have to write this number in the form 10 T plus Cute.
Now, if I do that, that means that tea is 103 on cue.
Is five now using your DV number?
What you do is you create.
So let's call the D V number.
Okay, create a new number.
Which is this thing.
Okay, so let's calculate what this is.
Okay, so em is seven.
Kira is five.
This is 103.
That happens to be 138 which hopefully you mental over to rip Music is good enough to check with at that.
See that?
That is a multiple of 23.
So if the number you end up with here is a multiple of the 23 then the number you're testing is a multiple of 23.
So you can adopt this method any number that ends in 137 or nine and it will work.
Okay.
What it relies on is that the number is constructed in this way.
It's guaranteed that if something divides this number, then it It will also divide this number.
And you can easily prove that not too difficult to prove not the easiest process to run through.
But now it's not that.
Then it's applying for any number.
But this is this is completely general thing.
Now it doesn't apply to numbers divisible by five bit, almost applies to all the odd numbers, the idea is that the goal is the challenge, of course, is to get you've got to get.
It's very important that you get to this point where you have something that ends in and nine.
That's what you need and then singled us out.
One.
Business.
Blood, blood, Blood.
Right?
So that's that's important, Of course, that works with the war on threes, sevens and nines that you could do that for.
Do you like that?
It's general.
It's not particularly.
I mean, come on.
Alcohol.
Good deal.
23 times Table Brady.
Exactly What do you look at that and think that that's a cute trick?
Or do you think that was a mess?
If you want to know what I like about it is.
So when I found out about it, a sort of immediately wanted to prove it.
Okay, I want to see why it works.
That's what that's what I think that's true.
Anybody who's into masses, okay, it works.
But why does it work?
Let's prove that you immediately sit down and prove it to yourself, and you understand why it works.
I think that's where what I liked when I saw White worked, which is in the proof.
So I think we should leave the viewers to prove it right.
Here we go.
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66 got worse card trick.
Well, you got patient audience.