 # Factors and Factoring

Factors are all about multiplication. Every number that you multiply is called a factor. 4*3=12 has the two factors 4 and 3. But what are all of the factors of twelve?

We already know about 4 and 3 (4 * 3 = 12).
Don't forget about 2*6=12. So 2 and 6 are also factors.
And then there's 1*12=12. So we finish with 1 and 12 being factors too.
Overall, the factors of 12 are 1, 2, 3, 4, 6, and 12.

If you look on a multiplication table you could have looked for the number twelve across the chart and also found all of the factors. 1*12, 12*1, 2*6, 6*2, 3*4, and 4*3 all can be multiplied to make 12. With factors, we are only looking for whole numbers that are equal to or less that the original number.

# Prime Numbers and Factoring

The official definition says a factor is a number that can be divided into another number with no remainder. Is 6 a factor of 6... yes. Is 6 a factor of 12, 18, and 24... yes. Is 6 a factor of 13? No. 13 divided by 6 is 2 with a remainder of 1. Wait a minute. What are the factors of 13?

1 * 13 = 13
13 * 1 = 13
Ummm. That's it.

Since 13 can only be divided evenly by 1 and 13 (itself), it is called a prime number. A prime number can only be divided evenly by one and itself. The other numbers such as 6, 12, and 21 are called composite numbers since they can have many factors (more than two). There are a lot of prime numbers and mathematicians still have contests to find the largest ones.

Prime Number Examples: 2, 3, 5, 7, 11, 13, 17, 23...

'One' is NOT a prime number. 1 is a very special number on its own

So we've got factors and prime numbers. In your classes you'll be asked to find prime factors. That's like the ultimate breakdown of factors into a series of prime numbers. Let's look at some examples.

Factors of 4: 1, 2, 4 (1*4, 2*2).
Prime factors of 4: 2, 2 (we had to drop the 1*4 because one is not prime).

Factors of 12: 1, 2. 3, 4, 6, 12.
Prime factors of 12: 2, 2, 3.
Prime factorization of 12: 2 * 2 * 3 = 12

# Greatest Common Factor

We're also going to talk about the greatest common factor. The GCF looks at the list of factors for two numbers and finds the largest one they have in common. What about 4 (1,2,4) and 6 (1,2,3,6)? When you look at the lists you can see the GCF is 2. There are larger values in the list, but those larger numbers are not in both lists. Here's an example.

Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 GCF: 6

Since we're here...
Prime factorization of 18: 2 * 3 * 3 = 3 * (2 * 3)
Prime factorization of 24: 2 * 2 * 2 * 3 = 2 * 2 * (2 * 3)

Do you see the six in there (2*3)? The common factors reveal the GCF when you look for the pattern. So there are two ways to find the value, common prime factors and common factors.

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