Factors, Prime Numbers & Prime Factorization

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. All About Primes Click to Advance

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 Click to Advance

Can You divide any even number by 2using 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 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 40 2 20 2 10 2 5 40= 2∙2∙2∙5 = 23∙5 All About Primes Click to Advance

Can You divide any number by 3using Shorthand Division? Will it divide evenly? 6+1+2+5+4=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 3. Divide 102,072 by 3. • It’s 14,228 It’s 34,024 All About Primes 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 3 • Collect the “dangling” numbers as a product, optionally using exponents 36 2 18 2 9 3 3 36= 2∙2∙3∙3 = 22∙32 All About Primes 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 5 • Collect the “dangling” numbers as a product 150 2 75 3 25 5 5 150 = 2∙3∙5∙5 All About Primes Click to Advance

What is a Prime Number? • A Whole Number is primeif it is greater than one, andthe 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 Primes Click to Advance

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=49 112=121 132=169 172=289 192=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 121 169 All About Primes Click to Advance

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= 17 because 17 is prime All About Primes Click 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 198 2 99 3 33 3 11 198=2·3·3·11 All About Primes 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 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 5 33 311 165=3·5·11 All About Primes Click to Advance

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

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· 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 Click to Advance

Factors, Prime Numbers & Prime Factorization