Transcript Prime Factorization

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 the
number by.
• If the only answers are the number itself
and 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
3x2
• For 6 we write the prime
factorization as
• 6=3x2
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.
• 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
2
3 6
2 12
2 24
• The prime factorization of 24 is 2 x 2 x 2 x 3
The advantages to this method are in the ease of choosing your
numbers. You must choose a prime number to divide by each
time, which limits your choices to a few small numbers usually.
The reason this is sometimes called the birthday cake method is
because of the resulting shape.
2
3 6
2 12
2 24
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
• First we divide 24 by 8. The result is 3.
i.e. 8 x 3 = 24
24
• Next we look at 8. We can divide 8 by 4 to
get a result of 2.
8
• 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.
4
• 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
2
3
2
2
• The advantage to the factor tree
method is the ability to choose
from any number. (as long you can
divide the original number evenly)
24
8
• This method is named for the
resemblance the final results
have to a pine tree.
4
2
3
2
2
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).