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2. Prime numbers & Prime Factorization
Factors & Multiples
Prime numbers & Prime Factorization
Copyright©2001 Lynda Greene

3. Prime Numbers
Copyright©2001 Lynda Greene

4. Examples: Let’s look at the factors of several numbers.
Before we learn to find the Prime Factors of a number, we need to know what a Prime Number is.
A Prime Number is a number that has only two FACTORS (numbers that will divide evenly with no remainders) These two factors are the original number and the number 1.
Examples: Let’s look at the factors of several numbers. 2 3 6 4 5 7
1 x 2
1 x 3
These two numbers (4 and 6) have more than
one set of factors, so they are called “composite numbers”
1 x 4
2 x 2
1 x 5
1 x 6
2 x 3
1 x 7
The other numbers (2, 3, 5, and 7) have only one set of factors each,
the number one (1) and the number itself.
This means that 2, 3, 5, and 7 are Prime Numbers!
Copyright©2001 Lynda Greene

5. To find the Prime Factorization of any number, you must divide over and over by bigger and bigger prime numbers. But before you can do that, you need to know which numbers are prime.
One way to do this is to have a list of prime numbers that you can refer to, but the easiest way (in the long run) is to memorize the prime numbers.
(It helps to know your Prime Numbers later when you learn to reduce fractions, simplify square roots and factor polynomials)
Memorizing tip #1:
Look at the last digits in the middle two rows, they are almost exactly the same.
(3rd row: last number changed, 4th row: has a number missing)
Here is a list of the first 25
prime numbers.
2, 3, 5, 7, 11, 13, 17, 19,
23, 29, 31, 37, 41, 43, 47,
53, 59, 61, 67, 71, 73, 79,
83, 89,
Memorizing tip #2:
2 primes in the 20’s,
2 in the 30’s,
3 in the 40’s,
(then it repeats)
2 in the 50’s,
2 in the 60’s,
3 in the 70’s,
(starts to repeat again)
2 in the 80’s
Grouping Pattern: two two three
20’s ’s ’s
50’s ’s ’s
80’s ...pattern breaks down
Copyright©2001 Lynda Greene

6. Practice Problems: (Hit enter to see the answers)
Label each number below as prime, composite or neither
1) ) 51
2) ) 23
3) ) 2
4) ) 1
Answers:
1) prime
2) composite
3) composite
4) prime
5) composite
6) prime
7) prime
8) neither
Copyright©2001 Lynda Greene

7. Prime Factorization
Copyright©2001 Lynda Greene

8. Prime Factorization: Breaking a number up into the smallest possible pieces. These pieces are called “prime factors” and they are a group of “prime numbers” that when multiplied together are equal to the original number.
Example: The prime factorization for 72 is:
2 x 2 x 2 x 3 x 3 or (23 x 32)
These expressions multiply together to give you 72 and
2 and 3 are both prime numbers.
Copyright©2001 Lynda Greene

9. Example: Find the Prime Factorization for the number 36
General Steps:
Divide the number by 2 (since 2 is the first prime number in the prime number list)
Divide the answer by 2
*Keep using 2 until it doesn’t divide evenly anymore*
3) Divide by 3, until it doesn’t divide evenly
Divide by 5, 7, 11, ... (each of the prime numbers)
STOP ! when you get a
Prime Number on the bottom.
Here is a list of the first 25
prime numbers.
2, 3, 5, 7, 11, 13, 17, 19,
23, 29, 31, 37, 41, 43, 47,
53, 59, 61, 67, 71, 73, 79,
83, 89,
Note: Many teachers insist on using this upside-down division symbol for Prime Factorization, but it is still plain old division, just write the answer underneath instead of on top. 2 ) 36
The answer is made up of the prime numbers on the outside of the division symbols. They must be written with multiplication signs between them.
Divide by 2 2 ) 18
Divide by 2
Can’t divide by 2
anymore, go to next
Prime Number (3) ) 3 9
Divide by 3 3
Prime Number
STOP!
Check: 2 x 2 x 3 x 3 = 36
You will get the original
number back if your answer is correct.
ANSWER: 2 x 2 x 3 x 3 or 22 x 32
Copyright©2001 Lynda Greene

10. Example: Find the Prime Factorization for the number 42
List of prime numbers.
2, 3, 5, 7, 11, 13, 17, 19,
23, 29, 31, 37, 41, 43, 47,
53, 59, 61, 67, 71, 73, 79,
83, 89,
Divide 42 by 2, then divide the answers by 2 until they won’t divide evenly anymore, then divide by the next prime number (3, 5, 7, 11,...). Stop when you get a Prime Number on the bottom.
ANSWER: 2 x 3 x 7 2 42 )
Divide by 2
Can’t divide by 2
anymore, go to next
Prime Number (3) 3 ) 21
Divide by 3 7
Check your answer,
2 x 3 x 7=42
Prime Number
STOP!
Copyright©2001 Lynda Greene

11. Example: Find the Prime Factorization for the number 126
List of prime numbers.
2, 3, 5, 7, 11, 13, 17, 19,
23, 29, 31, 37, 41, 43, 47,
53, 59, 61, 67, 71, 73, 79,
83, 89, 2
126 )
Divide by 2
Can’t divide by 2
anymore, go to next
Prime Number (3) 3 ) 63
Divide by 3 3 ) 21
Divide by 3
ANSWER: 2 x 3 x 3 x 7
or 2 x 32 x 7 7
Prime Number
STOP!
Check your answer,
2 x 3 x 3 x 7 = 126
Copyright©2001 Lynda Greene

12. Example: Find the Prime Factorization for the number 220
List of prime numbers.
2, 3, 5, 7, 11, 13, 17, 19,
23, 29, 31, 37, 41, 43, 47,
53, 59, 61, 67, 71, 73, 79,
83, 89, 2
220 )
Divide by 2 2
110 )
Can’t divide by 2 anymore, go to next
Prime Number (3)
Divide by 2 5 ) 55
Can’t divide by 3 either, go to next Prime Number (5)
Divide by 5 11
Prime Number
STOP!
ANSWER: 2 x 2 x 5 x 11
Check your answer,
2 x 2 x 5 x 11 = 220
Copyright©2001 Lynda Greene

13. Example: Find the Prime Factorization for the number 273
List of prime numbers.
2, 3, 5, 7, 11, 13, 17, 19,
23, 29, 31, 37, 41, 43, 47,
53, 59, 61, 67, 71, 73, 79,
83, 89,
Can’t divide by 2 at all,
go to next Prime Number (3) 3
273 )
Divide by 3 7 91 )
Can’t divide by 3 anymore, go to next Prime Number (5)
Divide by 7 13
Can’t divide by 5 either, go to next Prime Number (7)
Prime Number
STOP!
ANSWER: 3 x 7 x 13
Check your answer,
3 x 7 x 13 = 273
Copyright©2001 Lynda Greene

14. Practice Problems: (Hit enter to see the answers)
Find the prime factorization for the following numbers
Answers:
1) 3 x 5 x 7
2) 2 x 2 x 2 x 3 x 3
3) 3 x 3 x 5 x 5
4) 3 x 3 x 3 x 5
5) 2 x 3 x 3 x 5
6) 3 x 3 x 7
7) 2 x 7 x 11
8) 2 x 3 x 7 x 7 x 11
105
2) 72
225
4) 135 90
6) 63
7) 154
8) 3234
Copyright©2001 Lynda Greene

15. Questions? send e-mail to: lgreene1@satx.rr.com
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