Prime Factorization

No, not this kind of prime.

What is a prime?

• A prime is a natural number greater than 1 that has no positive divisors other than 1 and itself.

• In other words, see what you can divide thenumber by.

• If the only answers are the number itselfand one, then the number is prime.

And if it’s not prime?

If a number is not prime it is called composite.6 is divisible by 1, 2, 3, and 6. This means that 6 is a composite number.

So what is prime factorization?

• This is a way to break a number down into parts where each part is a prime number.

Let’s look at the number 6.

6 can be broken down into 3 x 2

6 3 x 2

• For 6 we write the prime factorization as

• 6 = 3 x 2

How do we find the prime factorization?

• Division by primes

• Factor Tree

Division by Primes

• The first method is called division by primes (or the birthday cake method)

• Let’s use the number 24• First we pick a prime that we can divide 24 by.

Since it is even we will try 2.

242122

632

• First we divide 24 by 2 and the result is 12

• Next we choose another prime to now divide 12 by

• We divide 12 by 2 and the result is 6

• Now we choose a prime to divide 6 by, let’s choose 3 this time

• The result is 2, which is a prime, once a prime answer is reached we stop

• Now to get our answer we look on the outside of the long division

• The prime factorization of 24 is 2 x 2 x 2 x 3

242122

632

The advantages to this method are in the ease of choosing yournumbers. You must choose a prime number to divide by eachtime, which limits your choices to a few small numbers usually.

The reason this is sometimes called the birthday cake method isbecause of the resulting shape.

Factor Tree

• The second method is called a factor tree• Again we will use the number 24• This time we are not limited to our choice of

numbers to divide by so we will pick 8

24

8 3

4 2

22

• First we divide 24 by 8. The result is 3. i.e. 8 x 3 = 24

• Next we look at 8. We can divide 8 by 4 to get a result of 2.

• Then we look at the 3. This is a prime so we do not touch the 3. We will circle it to indicate we are done with it.

• The 4 can turn into 2 x 2.

• The 2 is prime so we circle it.• The last two 2’s are prime, so again we circle them.

• Now look at all of our circled numbers. This is our prime factorization. 24 = 3 x 2 x 2 x 2

24

8 3

4 2

22

• The advantage to the factor tree method is the ability to choose from any number. (as long you can divide the original number evenly)

• This method is named for the resemblance the final results have to a pine tree.

Which method should we use?

• Both division by primes and factor tree may be used in any case; however, one method may make the problem much easier.

• Often it is best to use division by primes when it is difficult to see large numbers that can be used.

• Factor tree is a good choice when it is easy to see a large number to divide by. For example, numbers ending in zero (divide by ten).