< back to Selected Math Topics >

## The Math Tutor - Austin

How To Find the Factors of a Number

First off, you should know that a number can be factored in LOTS of different ways.

For example:

360 = 2*180 = 3*120 = 4*90 = 5*72 = 6*60 = 8*45 = 9*40 = 10*36 = 12*30 = 15*24 = 18*20 = ...

Wow! That's a lot of factor pairs. Isn't there any way we could get just one answer?

Prime Factorization

Factoring a number into a pair of factors has lots of answers, but - luckily there's a different way to factor that will give us only one answer

Instead of factoring the number into a pair, we're going to break it down as much as possible - that is, into factors which are prime numbers. (In case you don't remember, a prime number has no factors besides the obvious ones - i.e. "1" and the number itself).

The process is pretty simple - we're just going to break up any factors we find into smaller factors, until there's nothing left but primes.

Here's an example:

Find the prime factorization of 1890.

Well, first off, 1890 has a factor of 10 (because it ends in zero):

1890 = 189 * 10

Each of the remaining numbers - 189 and 10 - can be factored further.

For example, 10 can be written as 2*5, so we have:

1890 = 189*(2*5)

2 and 5 are both prime, so we're done with them. But - 189 still has a factor of 9 (because the digits 1+8+9 add up to 18, which is divisible by 9). If we divide out that 9, we find that 189 is 9*21, so we can write

1890 = (9*21)*2*5

And of course, 9 = 3*3 and 21 = 3*7:

1890 = (3*3)*(3*7)*2*5

Which leaves us nothing but primes - so we're done factoring!

Of course, we could clean it up a little and write the factors in order from smallest to largest:

1890 = 2*3*3*3*5*7

Finding bigger factors

Finally, if you were hunting for bigger factors, well - they're easy to construct from the prime factors.

Clearly any combination of prime factors will also be a factor of our original number.

Example:

1890 = 2*3*3*3*5*7

To make bigger factors, we'll just multiply some of the little factors together:

2*3 = 6 is a factor of 1890

3*3 = 9 " "

3*5 = 15

2*3*3 = 18

3*7 = 21

3*3*3 = 27

5*7 = 35

(etc....)