1. Prime Numbers and Prime Factorization

2. Factors
Factors are the numbers you multiply together to get a product.
For example, the product 24 has several factors.
24 = 1 x 24
24 = 2 x 12
24 = 3 x 8
24 = 4 x 6
SO, the factors are 1, 2, 3, 4, 6, 8, 12, 24

3. Finding Factors Start with 1 times the number. Try 2, 3, 4, etc.
When you repeat your factors, cross out the repeat - you’re done at this point.
If you get doubles (such as 4 x 4), then you’re done. Repeats or doubles let you know you’re done.

4. What are the factors of 16? 1 x 16 2 x 8 3 x ?? 4 x 4
3 is not a factor, so cross it out
4 x 4
doubles = done
The factors of 16 are 1,2,4,8,16

5. What are the factors of 18? 1 x 18 The factors are 1,2,3,6,9,18 2 x 9
Repeat! Cross it out! We’re done!

6. What are the factors of 7? The only factors of 7 are 1,7 1 x 7 2 x ??
This works, but it is a repeat. We are done.

7. Prime and Composite Numbers
Prime numbers are
numbers that only have
two factors: one, and the
number itself.
EXAMPLES:
3, 5, 7, 11, 31
Composite numbers
have more than two
factors.
EXAMPLES:
6, 15, 18, 30, 100

8. A Product of Primes
Every composite number can be expressed as a product of prime numbers.
This is called prime factorization.

9. It can be expressed as a product of primes: 3 x 5
Example
15 is a composite number.
It can be expressed as a product of primes: 3 x 5

10. To find the prime factorization:
Divide the number by the first prime number possible.
Circle the prime number, and continue with the other factor.
Divide the new factor by a prime number.
Continue this process until the only numbers you have left are prime numbers.

11. Remember the Prime Number List:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97…

12. Example: Prime Factorization of 100.
100 ÷ 2 = 50. Two is the first prime number that goes into 100.
Now we deal with the 50. Divide it by 2 to get the next factors.
2 is a prime number, so we are done with it.
2 X 50
25 is not divisible by the first prime, 2. The next prime, 3, does not work either. We must divide by 5 to get a factor.
2 X 25
Both numbers are prime, leaving us with all primes.
5 x 5

13. What’s the Answer?
Now, we just list our factors with multiplication signs between them. Use the circled prime numbers.
2 x 2 x 5 x 5
We have listed 100 as a product of prime numbers.

14. Exponent Form
We have just listed our prime factorization for 100 as being 2 x 2 x 5 x 5. This is repeated multiplication. Repeated multiplication can be expressed with exponents.
Our prime numbers are our bases. The number of times the prime number is written is the exponent.
2 x 2 can be expressed in exponent form: 22
5 x 5 can be expressed as 52
Put it together, and 2 x 2 x 5 x 5 is more simply put as
22 x 52

15. Another Example
420
2 x 210
2 x 105
22 x 3 x 5 x 7
3 x 35
5 x 7

16. Try this on your own: 54
Answer:
2 x 33

17. Homework Time!