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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. - PowerPoint PPT Presentation
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Factors, Prime Numbers & Prime Factorization
All About Primes 1Click to Advance
Suggestion:Work with scratch paper and pencil as you go through this presentation.
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 greaterthan 1 whose ONLY factors are 1 and itself.
Example: 7 is a Prime Numberbecause 7’s only factors are 1 and 7
How can you check to see if a number is Prime?
Tricks for recognizing when a numbermust have a factor of 2 or 5 or 3
ANY even number can always be divided by 2◦ Divides evenly: 3418, 70, 122◦ Doesn’t: 37, 120,001
Numbers ending in 5 or 0 can always be divided by 5◦ Divides evenly: 2345, 70, 41,415◦ Doesn’t: 37, 120,001
If the sum of a number’s digits divides evenly by 3, then the number always divides by 3◦ Divides evenly: 39, 186, 5670 ◦ Doesn’t: 43, 56,204
All About Primes 2Click to Advance
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 2. Divide 102,072 by 2. It’s 21,342 It’s 51,036
All About Primes 3
724,013458,026214
toindiv
12
Click to Advance
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
All About Primes 4
40
2 20
2 10
2 5
40= 2∙2∙2∙5 = 23∙5
Click to Advance
Can You divide any number by 3 using Shorthand Division?
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 3. Divide 102,072 by 3. It’s 14,228 It’s 34,024
All About Primes 5
814,02452,1632412
toindiv
2412 1212
Will it divide evenly?6+1+2+5+4=18, 18/3=6 yes
Click to Advance
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 3Collect the “dangling”
numbers as a product, optionally using exponents
All About Primes 6
36
2 18
2 9
3 3
36= 2∙2∙3∙3 = 22∙32
Click to Advance
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 5Collect the “dangling”
numbers as a product
All About Primes 7
150
2 75
3 25
5 5
150 = 2∙3∙5∙5
Click to Advance
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 numbers2 is the only even prime number
◦ For example, 18 = 2∙9 so 18 isn’t prime3, 5, 7 are primes 9 = 3∙3, so 9 is not prime 11, 13, 17, and 19 are primeThere are infinitely many primes above 20. How can you tell if a large number is prime?
All About Primes 8Click to Advance
Is a large number prime? You can find out!What smaller primes do you have to check?
See where the number fits in the table above
Let’s use 151 as an example:
151 is between the squares of 11 and 13Check 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 9
Here is a useful table of the squares of some small primes:22=4 32=9 52=25 72=49 112=121 132=169 172=289 192=361
421
4
911141517
r813
83341
1115111
r
Click to Advance
121 169
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 ∙ 370 = 2 ∙ 5 ∙ 724 = 2 ∙ 2 ∙ 2 ∙ 3 = 23 ∙ 317= 17 because 17 is prime
All About Primes 10Click to Advance
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
All About Primes 11
198
2 99
3 33
3 11
198= 2·3·3·11
Click to Advance
The mechanics ofThe “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 3333’s digits add up to 6,
so 3 must be a factorDivide 3 into 33 to get 11All the “dangling” numbers are
prime, so we are almost doneCollect the dangling primes into a
product (smallest-to-largest order)
All About Primes 12
165
5 33
3 11
165=3·5·11
Click to Advance
Thank YouFor Learning about Prime
Factorization
All About Primes 13
Press the ESC key to exit this Show
You can also use a linear approach
84=2· 42 =2· 2· 21 =2· 2· 3· 7 =22· 3· 7 (simplest form)216=2· 108 =2· 2· 54 =2· 2· 2· 27 =2· 2· 2· 3· 9 =2· 2· 2· 3· 3· 3 =23·33 (simplest form)
All About Primes 14
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.
1082162 54
108227542
9273
393
Click to Advance