 So one of the important things we need to be able to do in mathematics is to be able to factor and the bad news the hardest easy problem in mathematics is the problem of Factoring to rewrite an expression as the product of two or more expressions We say this is the hardest easy problem because it's very easy to explain what we want to do We want to write something as a product. No problem. The difficulty is that it's actually very hard to figure out what those factors are So let's start out with a couple of basic ideas. If n is a whole number We have the following definitions a whole number n is said to be composite if it can be written as a product of smaller whole numbers Now we can't always do that so a whole number n greater than one is said to be prime if it's not composite and An important idea here is that one is not considered prime or composite. It's a thing by itself So let's determine whether 100 is prime or composite Definitions are the whole of mathematics. All else is commentary. So let's pull in our definition of what composite and prime mean So the number is composite if I can write it as a product of smaller numbers and it's prime if I can't and so we observe that 100 can be written as a product of smaller numbers Actually, there's many ways of doing this. Maybe one of them is 10 times 10 And so 100 can be expressed as a product of smaller numbers. So 100 is composite Let's determine whether 5 is prime So with 100 it was actually pretty easy because it didn't take us a lot of effort to come up with two smaller numbers that multiplied to 100 but 5 is a little bit more difficult We have to check the product of smaller whole numbers and see if any of them multiply to five So the numbers smaller than five are one two three and four So we have to check every possible product of these numbers Well, maybe not one useful thing is that multiplication is commutative so that a times b is the same as b times a So if I check two times one, I don't need to check one times two So about half of these products. I don't need to worry about But we do need to check all the other products and We see that none of these is equal to five and since no product of smaller numbers is equal to five Five is prime Now that's a lot of work And so the first thing a mathematician asks when confronted with a problem That's a lot of work is is there an easier way and The answer is yes as long as we remember our definitions So remember definitions are the whole of mathematics and one of those important definitions is that of division a divided by b is equal to c if and only if a is equal to b times c and What this means is that if I can write a number as a product then I can do a division that comes out evenly So instead of seeing if two smaller numbers multiply to n we can see if n is divisible by a smaller number So we could try dividing five by the smaller numbers So we find five divided by one And our definition of division says that since five divided by one is five Then we know that five is equal to one times five and we've written five as a product Unfortunately, that doesn't work because we need to be able to write this as a product of smaller numbers and five is not smaller than five and so this division doesn't help since we have not written five as a product of smaller numbers But we can try other divisors. So five divided by two is and Since there's a remainder five is not two times something five divided by three is And again, there's a remainder. So we go on to five divided by four And we've run out of numbers that are smaller than five and so we can conclude as before Since five cannot be written as a product of smaller numbers. It is prime So let's look at those composite numbers. Suppose a number is not prime Definitions are the whole of mathematics So remember not being prime is the same as being composite and the composite number can be written as a product of smaller numbers So if our number is not prime, we can write our number n as the product a times b And we define factors as follows suppose n is equal to a times b Then we say that a and b are factors of n and we also say that n is a product of a and b And this leads us to an important idea a prime Factorization of a number n is a product equal to n where each factor is a prime number Now if I want to write a number as a product of primes I have to know what the prime numbers are and we've already determined that five is a prime number But what about others? And here's the bad news. There's no easy way to determine if a number is prime We have to go through some process of trial and error before we determine that a number is prime and it takes a lot of effort There aren't very many things that are worth memorizing in mathematics But one of the things that it is useful to remember are the primes at least the primes under 20 And so here they are So we might try to find a factorization of 30 and a prime factorization of 30 So again definitions are the whole of mathematics all else is commentary a factorization just means we want to write 30 as a product and so we might use 30 is equal to How about three times ten? So remember we also want to find a prime factorization And so remember definitions are the whole of mathematics all else is commentary a prime factorization is a factorization where all of our factors are prime numbers and Maybe we're fantastically lucky or maybe the universe is a kind and gentle place Don't count on it So we recall our list of prime numbers and three is a prime number But ten isn't And so this is not a prime factorization However, we do note that ten can also be written as a product ten is equal to two times five Equals means replaceable so any place I see ten I can replace it with two times five And so that means we can write 30 as the product three times two times five and Three two and five are all primes And so this is a prime factorization of 30