Prime and composite: the rectangle test for a number's factors

0:00Two friends are packing fruit into boxes. One has twelve figs. The other has seven. The rule is simple. The fruit has to sit in a neat rectangle, with every row the same length. The friend with twelve tries it, and finds she has choices. More than one rectangle works. The friend with seven tries it, and something strange happens. Before I show you, guess. How many different rectangles can you make from seven figs?

0:31Hold on to your number. That number is about to have a name. First, let's agree on what a rectangle is telling us. Here are twelve figs in three rows of four. Count across, and you get four. Count down, and you get three. Three rows of four is twelve. The two side lengths multiply to give the number. So every rectangle you can build is really a multiplication sentence, drawn instead of written.

1:02And one more rule, the same one the book uses. Turning a rectangle on its side does not make a new one. Three rows of four and four rows of three are the same shape, just rotated. We count it once. Now let's find every rectangle for twelve. One long line. Twelve in a single row. That's a rectangle, a very thin one. Two rows of six. That works, and it looks like a proper block.

1:33Three rows of four. That works too. What about five rows? Twelve doesn't split into five equal rows. One row would come up short. So twelve gives us exactly three rectangles. One by twelve, two by six, and three by four. Twelve has choices. Now seven. Two rows? Three and a half in each row. You'd have to cut a fig in half, so no. Three rows? Two in each row, and one fig left over, sitting outside the rectangle. No.

2:09Four, five, six rows? Every one of them leaves something over. The only thing that works is a single line of seven. So seven has exactly one rectangle. If you guessed one, you were right, and you have just met the whole idea of this video. Twelve has choices. Seven has none. Here's why that matters, and it's the neat part. Every rectangle hands you two numbers, and both of them are factors.

2:40The one-by-twelve rectangle hands over one and twelve. The two-by-six rectangle hands over two and six. The three-by-four rectangle hands over three and four. Stack them up. One, two, three, four, six, twelve. That is the complete factor list of twelve. And look what we didn't do. We never divided anything. We just drew boxes and read the sides. Three rectangles, six factors. Each rectangle is worth two. Seven's single rectangle hands over one and seven. That's it. Two factors, and no more.

3:21So now we can sort every number by counting its factors. A number with exactly two factors, just one and itself, is called a prime number. Seven is prime. So are two, three, five, eleven, thirteen, seventeen and nineteen. A number with more than two factors is called a composite number. Twelve is composite, with six factors. So are four, six, eight, nine, ten and fifteen. Notice that nine is on that list. Nine is odd, and nine is composite, because three times three is nine.

4:00Odd does not mean prime. That trips up more people than anything else in this chapter. Try the numbers from twenty-one to thirty yourself. Pause if you like. Only two of them are prime. Twenty-three and twenty-nine. Everything else in that decade breaks apart. Which leaves one number we've been quietly avoiding. The number one. One fig makes exactly one rectangle. A single square cell. So if you were counting rectangles, one would look just like seven. One rectangle each.

4:37But count factors instead, and they come apart. Seven's rectangle is one by seven. Two different side lengths, so two factors. One's rectangle is one by one. The two sides are the same number, so they collapse into a single factor. One has exactly one factor. Not two, and not more than two. So one is not prime, and one is not composite. It isn't a special case bolted on afterwards. It simply never enters either bucket.

5:14Now the number that surprises people. Two. Two is even. And two is prime. Its factors are one and two. That's exactly two factors, so it qualifies. Every other even number is composite, and the reason is easy to see. Four, six, eight, ten. Each one has two among its factors, and one, and itself. That's already three factors, at least. So two is the only even prime there will ever be.

5:45It's not an exception to the rule. It's the rule doing exactly what it says, which is counting factors, not checking whether a number is even. Let's test some numbers by hand. Twenty-three, fifty-one, thirty-seven and twenty-six. Twenty-six is quick. It's even, so two divides it, and twenty-six is two times thirteen. Composite. Twenty-three. Try two, no. Three, no. Four, no. And here you can stop. Here's a shortcut that isn't in the textbook, but it's true and it works.

6:23Factors come in pairs, and in every pair, one of them is small. Once the pairs cross over in the middle, you've already seen everything. For twenty-three, five times five is twenty-five, which is past twenty-three. So four was far enough. Twenty-three is prime. Thirty-seven, same story. Two, three, four, five, six, all fail, and seven times seven is forty-nine, past it. Thirty-seven is prime. Which leaves fifty-one, and this one is sneaky. It's odd. It doesn't end in five. It looks prime.

7:01But three times seventeen is fifty-one. Composite, hiding in plain sight. That's why we test instead of guess. Now five claims about primes. Some are true and some are not. Decide for yourself before I say. Claim one. No prime number ends in the digit four. That one is actually true. Anything ending in four is even, so two divides it. Claim two. Multiply two primes together and you get another prime.

7:35Two times three is six, and six is composite. Multiplying primes never gives a prime. Claim three. Primes have no factors. They have two. One, and themselves. A prime isn't a number you can't divide. It's a number with nothing else to divide by. Claim four. Every even number is composite. Two. Always two. It's the counterexample to half the wrong ideas about primes. Claim five. After two and three, you never get two primes right next to each other.

8:12That one's true, and here's why. Of any two numbers in a row, one is even, so once you're past two, one of them is out. Primes aren't just a list to memorise. They behave in patterns worth hunting for. Some primes come in twos, separated by a single number. Three and five. Seventeen and nineteen. We call those twin primes. Below one hundred there are eight such pairs, ending with seventy-one and seventy-three.

8:45Some primes stay prime when you write them backwards. Thirteen and thirty-one. Seventeen and seventy-one. Thirty-seven and seventy-three. Seventy-nine and ninety-seven. Here's one you can test right now. Take three digits, two, four and five, and make every three-digit number you can from them. Two four five. Two five four. Four two five. Four five two. Five two four. Five four two. Not one of them is prime, and you can see why without dividing at all. Every single one ends in two, four or five. So either two divides it, or five does.

9:29And a harder hunt. Which of forty-five, sixty, ninety-one, one hundred and five, and three hundred and thirty is built from exactly three different primes? Forty-five is three times three times five. Three primes, but a three appears twice. Sixty has an extra two. Ninety-one is only seven times thirteen. Three hundred and thirty runs to four different primes. The answer is one hundred and five. Three times five times seven. Three primes, all different, each used once.

10:08So what are primes actually for? Every composite number can be pulled apart into primes. That's what we just did with one hundred and five. And the primes can't be pulled apart any further. They're where the pulling stops. That's the real reason one is left out. If one counted as a prime, you could keep multiplying by one forever, and the pulling apart would never end. Primes are the building blocks. Every other number is something you can build out of them, and there's only one way to do it.

10:44That last part is a big claim, and we'll earn it properly in a later video. For now, here's a question to leave you with. The composite numbers thin out as you go up, and the primes get rarer and rarer. So do the primes eventually run out? Is there a biggest one? People wondered about that for a very long time. Tell me what you think in the comments, and we'll settle it soon.