# Factors, Prime Numbers

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Factors, Prime Numbers & Prime Factorization

All About Primes

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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 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?

Slide2Tricks for recognizing when a numbermust have a factor of 2 or 5 or 3

ANY even number can always be divided by 2Divides evenly: 3418, 70, 122 Doesn’t: 37, 120,001Numbers ending in 5 or 0 can always be divided by 5Divides evenly: 2345, 70, 41,415 Doesn’t: 37, 120,001If the sum of a number’s digits divides evenly by 3, then the number always divides by 3Divides evenly: 39, 186, 5670 Doesn’t: 43, 56,204

All About Primes

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Slide3Can 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 timeWhat’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

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Slide4Finding 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 factorsRepeat 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

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40 2 20 2 10 2 5 40= 2∙2∙2∙5 = 23∙5

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Slide5Can 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 timeDivide 3 into 6Goes 2 w/ no remainderDivide 3 into 1Goes 0 w/ 1 rem; carry it to the 2Divide 3 into 12Goes 4 w/ no remDivide 3 into 5Goes 1 w/ 2 rem; carry it to the 4Divide 3 into 24Goes 8 w/ 0 remYou try: Divide 42,684 by 3. Divide 102,072 by 3. It’s 14,228 It’s 34,024

All About Primes

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Will it divide evenly?

6+1+2+5+4=18, 18/3=6 yes

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Slide6Finding 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

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36 2 18 2 9 3 3 36= 2∙2∙3∙3 = 22∙32

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Slide7Finding 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

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150 2 75 3 25 5 5150 = 2∙3∙5∙5

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Slide8What 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 numberFor 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

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Slide9Is 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, 112 won’t work … 151 is not an even number3 won’t work … 151’s digits sum to 7, which isn’t divisible by 35 won’t work … 151 does not end in 5 or 07 won’t work … 151/7 has a remainder11 won’t work … 151/11 has a remainderSo … 151 must be prime

All About Primes

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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

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121 169

Slide10What 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

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Slide11Finding 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 pairsRepeat until every dangling number is primeCollect the “dangling” primes into a product

All About Primes

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198 2 99 3 33 3 11 198= 2·3·3·11

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Slide12The mechanics ofThe “Factor Tree” Method

First, find the easiest prime numberTo get the other factor, divide it into the original number2 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

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165 5 33 3 11 165=3·5·11

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Slide13Thank You

For Learning about Prime Factorization

All About Primes

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Press the

ESC

key to exit this Show

Slide14You 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

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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.

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Slide15
## Factors, Prime Numbers

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