Recognizing Prime Numbers Determine whether the following numbers are prime, composite, or neither. Just as a bit of a review, a prime number is a natural number, so one of the counting numbers 1, 2, 3, 4, 5, 6, and so on, that has exactly two factors. Its factors are 1 and itself. So an example of a prime number is 3. There's only two natural numbers that are divisible into 3: 1 and 3. Another way to think about it is the only way to get 3 as a product of other natural numbers is 1 × 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. First let's think about 24. Let's think about all of the natural numbers, or the whole numbers, although 0 is also included in the 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. Clearly, it is divisible by 1 and 24; in fact, 1 × 24 = 24. But it's also divisible by 2. 2 × 12 = 24, so it's also divisible by 12. It is also divisible by 3; 3 × 8 = 24. And 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. Let's just finish factoring it since we started it. It's also divisible by 4, and 4 × 6 = 24. So these are all of the factors of 24, clearly more than just 1 and 24. Now let's think about 2. The non-zero whole numbers that are divisible into 2 1 × 2 definitely works, 1 and 2, but there really aren't any others that are divisible into 2. So it has only 2 factors, 1 and itself. That's a definition of a prime number. So 2 is prime. 2 is prime. 2 is interesting, because it is the only even prime number. Only even prime number. And that might be common sense to 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, that's what makes it prime. But anything 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. Here is an interesting case: 1. 1 is only divisible by 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 it 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: 1, yourself, and some other things. So it's not composite. 1 is neither prime nor composite. 1 is neither. And finally we get to 17. 17 is divisible by 1 and 17. It's not divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, or 16. It has exactly two factors, 1 and itself, so 17 is prime.