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Video: The Definition of a Composite Number

Tim Burnham

What is a composite number?

03:14

Video Transcript

What is a composite number?

Let’s look at a couple of different definitions for a composite number. A composite number is a positive integer which is not prime (i.e., which has factors other than one and itself). Or a composite number is a number with more than two factors. Well we know from those two then that a composite number must be a positive integer. We know that it must have more than two factors. And that makes it not prime. And there are a couple of different definitions for prime numbers which you’ve probably heard of.

The first which is probably the more preferred definition is that it has exactly two factors. And the other which leaves a little bit of room for confusion, only one and itself are factors. And the slight confusion with the second one is what about the number one? One and itself one, yeah. One is a factor and one is a factor, so it sort of leaves that sort of area of confusion about whether or not one is a prime factor. But this definition here makes absolutely clear. One has only got one factor, one, so it’s not a prime number.

Okay let’s have a quick look at some examples then. I’ve just written out the positive integers from one to sixteen. And let’s identify what sort of number each of them are. Well one is not a prime number as we’ve just said, but it’s also not a composite number because it only has one factor. Two only has two factors, one and two, so that’s a prime number. Three also only has two factors, one and three, so that’s a prime number. Well one and four are factors of four, but two is also a factor of four, so it’s got more than two factors. It’s a composite number. Five has only got two factors, one and five, so that’s a prime number. Six has one and two and three and six as factors, so it’s a composite number. Seven has only one and seven as factors, so it’s a prime number. Eight has one and two and four and eight as factors, so that’s more than two. It’s a composite number. Nine has one and three and nine as factors. Well that’s more than two, so it’s a composite number. Ten has one and two and five and ten. That’s more than two, so it’s a composite number.

Eleven has only got two factors, one and eleven, so that’s a prime number. Twelve has got one and two and three and four and six and twelve as factors, so that’s a composite number. Thirteen’s got only two factors, one and thirteen, so that’s prime. Fourteen has got one and two and seven and fourteen as factors, so that’s a composite number. Fifteen has got one and three and five and fifteen. That’s more than two factors, so that’s composite. Sixteen has got one, two, four, and eight, and sixteen as factors. That’s more than two, so it’s also a composite number. So one is neither prime nor composite, but all the other positive integers are either prime or composite depending on how many factors they’ve got.