1. 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?
Suggestion:
Work with scratch paper and pencil as you go through this presentation.
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All About Primes

2. 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: , ,
Doesn’t: , ,001
Numbers ending in 5 or 0 can always be divided by 5
Divides evenly: , , ,415
Doesn’t: , ,001
If the sum of a number’s digits divides evenly by 3, then the number always divides by 3
Divides evenly: , ,
Doesn’t: , ,204
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All About Primes

3. 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
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?
You try: Divide 42,684 by Divide 102,072 by 2.
It’s 21, It’s 51,036
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All About Primes

4. Finding all factors of 2 in any number: The “Factor Tree” Method
Write down the even number
Break it into a pair of factors (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 40
40= 2∙2∙2∙5 = 23∙5
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All About Primes

5. Can You divide any number by 3 using Shorthand Division?
Will it divide evenly?
=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
Goes 2 w/ no remainder
Divide 3 into 1
Goes 0 w/ 1 rem; carry it to the 2
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
You try: Divide 42,684 by Divide 102,072 by 3.
It’s 14, It’s 34,024
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All About Primes

6. 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 36
36= 2∙2∙3∙3 = 22∙32
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All About Primes

7. Break 150 into a pair of factors (start with 2 and 75)
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
150
150 = 2∙3∙5∙5
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All About Primes

8. What is a Prime Number? 0 and 1 are not considered prime numbers
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?
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All About Primes

9. Is a large number prime. You can find out
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= = = = =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
7 won’t work … 151/7 has a remainder
11 won’t work … 151/11 has a remainder
So … 151 must be prime
All About Primes
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10. 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= because 17 is prime
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All About Primes

11. Finding all prime factors: The “Factor Tree” Method
Write down a number
Break it into a pair of factors (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
198
198= 2·3·3·11
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All About Primes

12. 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)
165
165=3·5·11
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All About Primes

13. Thank You For Learning about Prime Factorization
Press the ESC key to exit this Show
All About Primes

14. You can also use a linear approach
Suggestion:
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.
84=2· 42
=2· 2· 21
=2· 2· 3· 7
=22· 3· (simplest form)
216=2· 108
=2· 2· 54
=2· 2· 2· 27
=2· 2· 2· 3· 9
=2· 2· 2· 3· 3· 3
=23· (simplest form)
All About Primes
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