Recognizing prime and composite numbers

Determine whether the following numbers are prime, composite, or neither. So just as a bit of review, a prime number is a natural number-- so one of the counting numbers, 1, 2, 3, 4, 5, 6, so on and so forth-- that has exactly two factors. So its factors are 1 and itself. So an example of a prime factor is 3. There's only two natural numbers that are divisible into 3-- 1 and 3. Or another way to think about it is, the only way to get 3 as a product of other natural numbers is 1 times 3. So it only has 1 and itself. A composite number is a natural number that has more than just 1 and itself as factors. And we'll see examples of that and neither-- we'll see an interesting case of that in this problem. So first let's think about 24. So let's think about all of the-- I guess you could think of it as the natural numbers or the whole numbers, although 0 is also included in whole numbers. Let's think of all of the natural counting numbers that we can actually divide into 24 without having any remainder. We'd consider those the factors. Well, clearly it is divisible by 1 and 24. In fact, 1 times 24 is equal to 24. But it's also divisible by 2. 2 times 12 is 24. So it's also divisible by 12. And it is also divisible by 3. 3 times 8 is also equal to 24. And even at this point, we don't actually have to find all of the factors to realize that it's not prime. It clearly has more factors than just 1 and itself. So then it is clearly going to be composite. This is going to be composite. Now, let's just finish factoring it just since we started it. It's also divisible by 4. And 4 times 6-- had just enough space to do that. 4 times 6 is also 24. So these are all of the factors of 24, clearly more than just one and 24. Now let's think about 2. Well, the non-zero whole numbers that are divisible into 2, well, 1 times 2 definitely works, 1 and 2. But there really aren't any others that are divisible into 2. And so it only has two factors, 1 and itself, and that's the definition of a prime number. So 2 is prime. And 2 is interesting because it is the only even prime number. And that might be common sense you. Because by definition, an even number is divisible by 2. So 2 is clearly divisible by 2. That's what makes it even. But it's only divisible by 2 and 1. So that's what makes it prime. But anything else that's even is going to be divisible by 1, itself, and 2. Any other number that is even is going to be divisible by 1, itself, and 2. So by definition, it's going to have 1 and itself and something else. So it's going to be composite. So 2 is prime. Every other even number other than 2 is composite. Now, here is an interesting case. 1-- 1 is only divisible by 1. So it is not prime, technically, because it only has 1 as a factor. It does not have two factors. 1 is itself. But in order to be prime, you have to have exactly two factors. 1 has only one factor. In order to be composite, you have to have more than two factors. You have to have 1, yourself, and some other things. So it's not composite. So 1 is neither prime nor composite. And then finally we get to 17. 17 Is divisible by 1 and 17. It's not divisible by 2, not divisible by 3, 4, 5, 6. 7, 8, 9 10, 11, 12, 13, 14, 15, or 16. So it has exactly two factors-- 1 and itself. So 17 is once again-- 17 is prime.