Transcript Prime Factors

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Factors &
Multiples
Prime numbers &
prime factorization
Copyright©2001 Lynda Greene
Prime
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 can only be divided evenly (no
remainders) by itself and the number 1.
Examples: Let’s look at the factors of several numbers.
2
3
4
5
6
7
1x2
1x3
1x4
2x2
1x5
1x6
2x3
1x7
These two numbers (4 and 6) have more than
one set of factors, so they are called “composite numbers”
The other numbers (2, 3, 5, and 7) have only one set of factors each,
the number one (1) and itself (2, 3, 5, or 7).
These are Prime Numbers!
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,
3 29,
9 31,
1 37,
7 41,
1 43,
3 47,
7
53,
3 59,
9 61,
1 67,
7 71,
1 73,
3 79,
9
83,
3 89,
9 1 977
Grouping Pattern: two
two
three
20’s
30’s
40’s
50’s
60’s
70’s
80’s ...pattern breaks down
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
Practice Problems: (Hit enter to see the answers)
Label each number below as prime, composite or neither
1) 73
5) 51
2) 87
6) 23
3) 77
7) 2
4) 29
8) 1
Answers:
1) prime
2) composite
3) composite
4) prime
5) composite
6) prime
7) prime
8) neither
Prime
Factorization
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.
Example: Find the Prime Factorization for the number 36
Steps:
1)
2)
Divide 36 by 2 (the first prime number)
Divide the answer by 2
3)
4)
Divide by 3, until it doesn’t divide evenly
Divide by 5, 7, 11, ... (each of the prime numbers)
*Keep using 2 until it doesn’t divide evenly anymore*
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,
97
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.
Divide by 2
Divide by 2
Divide by 3
2 )36
2 )18
3) 9
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.
Can’t divide by 2
anymore, go to next
Prime Number (3)
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.
ANSWER: 2 x 2 x 3 x 3 or 22 x 32
Example: Find the Prime Factorization for the number 42
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.
Divide by 2
Divide by 3
2 )42
3 )21
7
Prime Number
STOP!
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,
97
ANSWER: 2 x 3 x 7
Can’t divide by 2
anymore, go to next
Prime Number (3)
Check your answer,
2 x 3 x 7=42
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,
97
Divide by 2
Divide by 3
2 )126
3 ) 63
3 )21
7
Divide by 3
Prime Number
STOP!
Can’t divide by 2
anymore, go to next
Prime Number (3)
ANSWER: 2 x 3 x 3 x 7
or 2 x 32 x 7
Check your answer,
2 x 3 x 3 x 7 = 126
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,
97
Divide by 2
Divide by 2
2 )220
2 )110
5 )55
11
Divide by 5
Prime Number
STOP!
Can’t divide by 2 anymore, go to next
Prime Number (3)
Can’t divide by 3 either, go to
next Prime Number (5)
ANSWER: 2 x 2 x 5 x 11
Check your answer,
2 x 2 x 5 x 11 = 220
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,
97
Divide by 3
Divide by 7
Prime Number
STOP!
3 )273
7 ) 91
13
Can’t divide by 2 at all,
go to next Prime Number (3)
Can’t divide by 3 anymore, go to
next Prime Number (5)
Can’t divide by 5 either, go to
next Prime Number (7)
ANSWER: 3 x 7 x 13
Check your answer,
3 x 7 x 13 = 273
Practice Problems: (Hit enter to see the answers)
Find the prime factorization for the following numbers
1) 105
2) 72
3) 225
4) 135
5) 90
6) 63
7) 154
8) 3234
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
End of Tutorial
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