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

724,013

458,026214

toindiv

12

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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

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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

814,02

452,1632412

toindiv

2412 1212

Will it divide evenly?6+1+2+5+4=18, 18/3=6 yes

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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

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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

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What is a Prime Number?

A Whole Number is prime if it is greater than one, andthe 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

9

11

14

1517

r813

8

33

41

11

15111

r

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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

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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

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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

1082

27542

9273

393

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