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Video: The Surprising Divisibility of Numbers

Tim Burnham

In this video we look at the divisibility of positive counting numbers, and investigate why it seems that so many 6 digit numbers are divisible by 143.

09:06

Video Transcript

In this video, we’re gonna look at the divisibility of numbers and find a couple of surprising results. Let’s just think about the positive counting numbers: one, two, three, four, five, and so on. These are called the natural numbers, and we’ve even got this special symbol, which represents natural numbers. Now, the properties that we’re gonna look at in this video also work on negative counting numbers, but let’s not worry about that for now.

What proportion of the natural numbers are divisible by two? Well, every other number is a multiple of two: two, four, six, eight, 10, and so on. So a half of the natural numbers are divisible by two. If you were to pick a natural number at random, half the time you’d pick one that’s divisible by two and the other half of the time you’d pick one that’s not divisible by two.

Okay, let’s think about what proportion of numbers are divisible by three. Well, every third number is a multiple of three. So a third of numbers are divisible by three. If I asked everyone watching this video to pick a natural number at random, a third of you would have picked a number that’s divisible by three. And we can also see that a quarter of the natural numbers are divisible by four, a fifth of them are divisible by five, and so on.

So let’s try an experiment. To be honest, this works best if you’re in a group of 30 or more people, and maybe some of you are — in a classroom for example. You’re going to need a calculator, so if you haven’t got one, pause the video now and go and get one. Right! So I want you to choose a random three-digit number, for example, one two three; although I’m sure you’ll come up with something much more imaginative than that. Now type it into your calculator. Now to make this a bit more interesting with bigger numbers, I want you to repeat those three digits to make a six-digit number. So my one two three becomes one two three one two three, for example.

And now, we’ve got a random six-digit natural number. Now, half of you should have a number that’s divisible by two, a third of you have a number that’s divisible by three, a quarter of you have a number that’s divisible by four, and so on. But what proportion of you will have a number that’s divisible by seven? Is it a seventh? Well, try dividing your six-digit number by seven. Is the answer a whole number? I bet it is for all of you. So that proportion is one, a 100 percent all of you have a number that is divisible by seven.

Okay, clear your calculator and type in that six-digit number again. What proportion of you have got a number that’s divisible by 91? So divide that number by 91 and see if you get a whole number answer. You’d think the answer would be a 91th, about one percent, but the proportion is one, a 100 percent. All of you should have a number that’s divisible by 91.

Right, one last go, clear your calculator; type in that six-digit number again. And the new question is what proportion of you have a number that’s divisible by 143? Well, it should be 143rd of you, just over half of one percent. But divide that number by 143 and I think all of you will get a whole number answer. The proportion is one, a 100 percent, all of you. So why are our right divisibility proportions breaking down? It’s weird, eh? Well, maybe not.

Let’s think about what happened when I told you to repeat your random three digits to make a new six-digit random number. So we start off with a three-digit number. In my case, one was in the 100s column, two was in the 10s column, and three was in the ones column. If I multiply that by 1000, I get 123000. 1000 times 100 is 100000, 1000 times 10 is 10000, and 1000 times one is 1000. So all those digits have moved to columns who have a place value 1000 times bigger.

Now, if I add my original number again, I get 123123. So what I’ve done is taken my original number, multiplied it by 1000, and then added another one. I’ve got 1001 of those numbers. What I’ve done is multiplying my original number by 1001. This means that the innocent sounding repeat those digits actually means multiply your number by 1001.

Now, let’s take a quick diversion. You’ve probably heard of prime numbers; they’re natural numbers that have got exactly two factors. You may have used the definition that they’re numbers, which are divisible by only one and themselves, but be careful because one isn’t a prime number because it’s only got one factor — one. So the prime numbers are two, three, five, seven, 11, 13, 17, 19, 23, 29, 31, 37, 41, and so on.

Now, you may be surprised to learn that all natural numbers greater than one can be expressed as the product of some prime numbers. For example, 210, and two times 105 equals 210 and two is a prime number remember. And 105 can be expressed as three times 35 and three is a prime number. And 35 is five times seven, and both five and seven are prime numbers. So two times three times five times seven is 210. And we call two times three times five times seven the product of primes for 210.

Now the slightly surprising thing is no matter what number you start off with, you can always find a bunch of prime numbers that will multiply together to make that number so long as you start off with a whole number greater than one. Let’s do the same for 1001. Well, seven times 143 is 1001, and seven is a prime number. And 11 times 13 is 143, and they’re also both prime numbers. So seven times 11 times 13 equals 1001; that’s the product of primes for 1001.

Now if we think back to our six-digit number, which is 1001 times our three-digit number, we can write it out like this. With 1001 being equal to seven times 11 times 13, we can replace the 1001 with seven times 11 times 13. Now, hopefully, we can see that our six-digit number is definitely divisible by seven because it’s seven times whatever this bit is.

So that’s how I knew all of your numbers were going to be divisible by seven. But multiplication is commutative, which is just a fancy way of saying that we’ll get the same answer, no matter which order we multiply our numbers together in. So instead of writing seven times 11 times 13 times 123, I could write seven times 13 times 11 times 123. And since seven times 13 is equal to 91, I knew your six-digit number was equivalent to 91 times something. In other words, it was divisible by 91 or I could multiply the 13 and the 11 together to get 143, and that told me that your number was divisible by 143.

So using a little bit of math knowledge and analytical skills, we’ve cleared up our little mystery. By thinking carefully about what mathematical operations we needed to apply to get to the six-digit number from our original three-digit random number, we could see that we were just multiplying it by 1001. And by using the fact that all natural numbers greater than one can be expressed as a product of primes, we can say that 1001 is equivalent to seven times 11 times 13. And by combining those prime factors in different ways, we can see that our six-digit numbers would all be divisible by seven, 11, 13, and seven times 11, so 77, and seven times 13, so 91, and 11 times 13, so 143, and of course seven times 11 times 13, which is 1001.

Then I carefully chose those numbers for you to test the divisibility, so I knew they’d all be factors of all of your numbers. If I picked any other factors to test, then the normal divisibility proportions would have applied. If I picked 12, then only a 12th of you would have had numbers divisible by 12. If I picked 142, then only 142th of you would have had numbers divisible by 142. So this surprising divisibility isn’t so surprising after all.