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CCSS.Math:

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 its factors are 1 and itself one and itself so with an example of a prime factor is 3 there's only two natural numbers that are divisible into 3 1 & 3 another way to think about it is the only way to get 3 is a product of other natural numbers is 1 times 3 so it only has 1 and itself a composite 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 all those 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 it's also divisible by 2 2 times 12 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 find 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 pause it 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 1 and 24 and now let's think about 2 well the whole the nonzero whole numbers that are divisible into two well one times two definitely works one and two but there really aren't any others that are divisible into two and so it only has two factors 1 and itself and that's the definition of a prime number so 2 is prime 2 is 2 is prime and 2 is interesting because it's the only it is the only even prime number only even only even prime number and that might be common sense to you because by definition an even number is divisible by 2 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 divisible that's even is going to be divisible by 1 itself itself and to any other number that is even it's going to be divisible by 1 itself and 2 so by definition it's going to have 1 in 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's an interesting here's an interesting case 1 1 is only divisible by 1 so it is not prime technically because it is not it only has 1 it only has 1 as a factor it does not have two factors one is itself but in order to be prime you have to have exactly two factors one has only one factor in order to be composite you have to have more than two factors you have to have one yourself and some other things so it's not composite so one is neither prime nor composite one is neither and then finally we get to 17 17 is divisible 17 is divisible by 1 and 17 they're just not divisible by 2 not difficult by 3 4 5 6 7 8 9 10 11 12 13 14 15 or 16 so it is it has exactly 2 factors 1 and itself so 17 is once again 17 is prime