Prime Factorization
Download
Report
Transcript Prime Factorization
Factors, Prime Numbers
& Prime Factorization
The Factors of a Whole Number are:
All the whole numbers that divide evenly into it.
Example: Factors of 12 are 1, 2, 3, 4, 6, and 12
Prime Numbers are any Whole Number greater
than 1 whose ONLY factors are 1 and itself.
Example: 7 is a Prime Number
because 7’s only factors are 1 and 7
How can you check
to see if a number is Prime?
Click to Advance
Suggestion:
Work with
scratch paper
and pencil as you
go through this
presentation.
All About Primes
1
Tricks for recognizing when a number
must have a factor of 2 or 5 or 3
ANY even number can always be divided by 2
◦ Divides evenly: 3418,
◦
Doesn’t:
37,
Numbers ending in 5 or 0 can always be divided by 5
◦ Divides evenly: 2345,
◦
Doesn’t:
37,
70, 122
120,001
70,
41,415
120,001
If the sum of a number’s digits divides evenly by 3, then
the number always divides by 3
◦ Divides evenly:
◦
Doesn’t:
39,
186, 5670
43, 56,204
Click to Advance
All About Primes
2
Can You divide any even number by 2
using Shorthand Division?
Let’s try an easy one. Divide 620,854 by 2:
Start from the left,
do one digit at a time
◦
◦
◦
◦
◦
◦
◦
14
What’s ½ of 6?
What’s ½ of 2?
What’s ½ of 0?
What’s ½ of 8?
What’s ½ of 5?
(It’s 2 with 1 left over; carry 1 to the 4, making it 14)
What’s ½ of 14?
div 2 in to 6 2 0, 8 5 4
3 1 0, 4 2 7
You try: Divide 42,684 by 2.
It’s 21,342
12
Divide 102,072 by 2.
It’s 51,036
Click to Advance
All About Primes
3
Finding all factors of 2 in any number:
The “Factor Tree” Method
Write down the even number
Break it into a pair of factors
40
(use 2 and ½ of 40)
As long as the righthand
number is even, break out
another pair of factors
Repeat until the righthand
number is odd (no more 2’s)
Collect the “dangling” numbers
as a product;
You can also use exponents
Click to Advance
2
20
2
10
2
5
40= 2∙2∙2∙5 = 23∙5
All About Primes
4
Can You divide any number by 3
using Shorthand Division?
Will it divide evenly?
6+1+2+5+4=18, 18/3=6 yes
Let’s try an easy one. Divide 61,254 by 3:
Start from the left,
do one digit at a time
◦ Divide 3 into 6
12
Goes 2 w/ no remainder
◦ Divide 3 into 1
24
div 3 in to 6 1, 2 5 4
Goes 0 w/ 1 rem; carry it to the 2
2 0, 4 18
◦ Divide 3 into 12
Goes 4 w/ no rem
◦ Divide 3 into 5
Goes 1 w/ 2 rem; carry it to the 4
◦ Divide 3 into 24
Goes 8 w/ 0 rem
12
24
You try: Divide 42,684 by 3.
It’s 14,228
12
12
Divide 102,072 by 3.
It’s 34,024
Click to Advance
All About Primes
5
Finding all factors of 2 and 3 in any number:
The “Factor Tree” Method
Write down the number
Break 36 into a pair of
factors (start with 2 and 18)
Break 18 into a pair of
factors (2 and 9)
9 has two factors of 3
Collect the “dangling”
numbers as a product,
optionally using exponents
Click to Advance
36
2
18
2
9
3
3
36= 2∙2∙3∙3 = 22∙32
All About Primes
6
Finding all factors of 2, 3 and 5 in a number:
The “Factor Tree” Method
Write down the number
Break 150 into a pair of
factors (start with 2 and 75)
Break 75 into a pair of
factors (3 and 25)
25 has two factors of 5
Collect the “dangling”
numbers as a product
Click to Advance
150
2
75
3
25
5
5
150 = 2∙3∙5∙5
All About Primes
7
What is a Prime Number?
A Whole Number is prime if it is greater than one, and
the only possible factors are one and the Whole Number itself.
0 and 1 are not considered prime numbers
2 is the only even prime number
◦ For example, 18 = 2∙9 so 18 isn’t prime
3, 5, 7 are primes
9 = 3∙3, so 9 is not prime
11, 13, 17, and 19 are prime
There are infinitely many primes above 20.
How
can you tell if a large number is prime?
Click to Advance
All About Primes
8
Is a large number prime? You can find out!
What smaller primes do you have to check?
Here is a useful table of the squares of some small primes:
22=4 32=9 52=25 72=49 112=121
121 132=169
169 172=289 192=361
See where the number fits in the table above
Let’s use 151 as an example:
151 is between the squares of 11 and 13
Check all primes before 13: 2, 3, 5, 7, 11
◦
◦
◦
◦
◦
2 won’t work … 151 is not an even number
3 won’t work … 151’s digits sum to 7, which isn’t divisible by 3
5 won’t work … 151 does not end in 5 or 0
13
21
11 151
7 won’t work … 151/7 has a remainder
7 151
11
14
11 won’t work … 151/11 has a remainder
41
11
r8
r4
So … 151 must be prime
Click to Advance
All About Primes
9
33
4
8
9
What is Prime Factorization?
It’s
a Critical Skill!
(A big name for a simple process …)
Writing a number as the product of it’s prime
factors.
Examples:
6=2∙3
70 = 2 ∙ 5 ∙ 7
24 = 2 ∙ 2 ∙ 2 ∙ 3 = 23 ∙ 3
17= 17 because 17 is prime
Click to Advance
All About Primes
10
Finding all prime factors:
The “Factor Tree” Method
Write down a number
Break it into a pair of factors
198
(use the smallest prime)
Try to break each new factor
into pairs
Repeat until every dangling
number is prime
Collect the “dangling” primes
into a product
2
99
3
33
3
11
198= 2·3·3·11
Click to Advance
All About Primes
11
The mechanics of
The “Factor Tree” Method
First, find the easiest prime number
To get the other factor, divide it
into the original number
2 can’t be a factor, but 5 must be
(because 165 ends with 5)
Divide 5 into 165 to get 33
33’s digits add up to 6,
so 3 must be a factor
Divide 3 into 33 to get 11
All the “dangling” numbers are
prime, so we are almost done
Collect the dangling primes into a
product (smallest-to-largest order)
Click to Advance
165
5
33
3
11
165=3·5·11
All About Primes
12
Thank You
For Learning about Prime Factorization
Press the ESC key to exit this Show
All About Primes
13
You can also use a linear approach
Suggestion:
84=2· 42
=2· 2· 21
=2· 2· 3· 7
=22· 3· 7 (simplest form)
If you are unable to do
divisions in your head,
do your divisions in a
work area to the right
of the linear
factorization steps.
216=2· 108
108
=2· 2· 54
2 216
=2· 2· 2· 27
9
=2· 2· 2· 3· 9
3 27
=2· 2· 2· 3· 3· 3
=23·33
(simplest form)
Click to Advance
54
2 108
All About Primes
27
2 54
3
39
14