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Factors, Prime Numbers & Prime Factorization All About Primes1 Click 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 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?

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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, 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 Primes2 Click to Advance

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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 Primes3 Click to Advance

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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 Primes4 40 2 20 2 10 2 5 40= 2∙2∙2∙5 = 2 3 ∙5 Click to Advance

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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 Primes5 Will it divide evenly? 6+1+2+5+4=18, 18/3=6 yes Click to Advance

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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 3 Collect the “dangling” numbers as a product, optionally using exponents All About Primes6 36 2 18 2 9 3 3 36= 2∙2∙3∙3 = 2 2 ∙3 2 Click to Advance

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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 5 Collect the “dangling” numbers as a product All About Primes7 150 2 75 3 25 5 5 150 = 2∙3∙5∙5 Click to Advance

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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? All About Primes8 Click to Advance

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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 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 Primes9 Here is a useful table of the squares of some small primes: 2 2 =4 3 2 =9 5 2 =25 7 2 =49 11 2 =121 13 2 =169 17 2 =289 19 2 =361 Click to Advance 121 169

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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 = 2 3 ∙ 3 17= 17 because 17 is prime All About Primes10 Click to Advance

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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 Primes11 198 2 99 3 33 3 11 198= 2·3·3·11 Click to Advance

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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) All About Primes12 165 5 33 3 11 165=3·5·11 Click to Advance

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Thank You For Learning about Prime Factorization All About Primes13 Press the ESC key to exit this Show

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You can also use a linear approach 84=2· 42 =2· 2· 21 =2· 2· 3· 7 =2 2 · 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 =2 3 ·3 3 (simplest form) All About Primes14 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. Click to Advance

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