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What is a composite number?
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Let’s look at a couple of different definitions for a composite number.
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A composite number is a positive integer which is not prime (i.e., which has factors other than one and itself).
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Or a composite number is a number with more than two factors.
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Well we know from those two then that a composite number must be a positive integer.
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We know that it must have more than two factors.
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And that makes it not prime.
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And there are a couple of different definitions for prime numbers which you’ve probably heard of.
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The first which is probably the more preferred definition is that it has exactly two factors.
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And the other which leaves a little bit of room for confusion, only one and itself are factors.
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And the slight confusion with the second one is what about the number one?
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One and itself one, yeah.
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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.
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But this definition here makes absolutely clear.
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One has only got one factor, one, so it’s not a prime number.
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Okay let’s have a quick look at some examples then.
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I’ve just written out the positive integers from one to sixteen.
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And let’s identify what sort of number each of them are.
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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.
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Two only has two factors, one and two, so that’s a prime number.
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Three also only has two factors, one and three, so that’s a prime number.
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Well one and four are factors of four, but two is also a factor of four, so it’s got more than two factors.
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It’s a composite number.
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Five has only got two factors, one and five, so that’s a prime number.
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Six has one and two and three and six as factors, so it’s a composite number.
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Seven has only one and seven as factors, so it’s a prime number.
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Eight has one and two and four and eight as factors, so that’s more than two.
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It’s a composite number.
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Nine has one and three and nine as factors.
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Well that’s more than two, so it’s a composite number.
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Ten has one and two and five and ten.
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That’s more than two, so it’s a composite number.
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Eleven has only got two factors, one and eleven, so that’s a prime number.
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Twelve has got one and two and three and four and six and twelve as factors, so that’s a composite number.
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Thirteen’s got only two factors, one and thirteen, so that’s prime.
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Fourteen has got one and two and seven and fourteen as factors, so that’s a composite number.
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Fifteen has got one and three and five and fifteen.
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That’s more than two factors, so that’s composite.
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Sixteen has got one, two, four, and eight, and sixteen as factors.
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That’s more than two, so it’s also a composite number.
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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.