Lesson 9 Prime Factorization - Welcome to the Home Page ...voyager.dvc.edu/~lmonth/PreAlg/lesson9student.pdf · Lesson 9 Prime Factorization Some whole numbers, such as 31, have only

Free Pre-Algebra Lesson 9 page 1

© 2010 Cheryl Wilcox

Lesson 9

Prime Factorization

Some whole numbers, such as 31, have only one pair of factors. The only way to write 31 as a product of whole numbers is 1 • 31. These numbers are called prime. Numbers with more than one pair of factors, such as 32, are called composite. Zero (0) and one (1) are not classified – they are neither prime nor composite. You might think that as the numbers get large, they would have more factors, and eventually we’d run out of primes. This isn’t true – there are infinitely many prime numbers. Euclid proved (circa 300 B.C.) that whatever prime we think is largest can be used to produce one still larger. It turns out that this distinction between prime and composite numbers is fundamental in understanding numbers. It’s strange but profound that some people (think Rain Man) can “see” whether or not a number is prime, or determine factors of large numbers without calculation. In the film Contact, an alien civilization communicates with primes (because whether or not a number is prime is not dependent on a specific number system), a communication received in the film by the SETI (Search for Extraterrestrial Intelligence) project. Primes have fascinated mathematicians for thousands of years. Most recently they have a new practical use in encrypting computer transmissions. Prime or Not? To determine whether a specific number is prime we look for factors, just as we did in the last lesson. If there are no factors other than the number itself and 1, the number is prime.

Example: Is 897 prime?

You don’t have to find all the factor pairs – just finding one pair other than 1 • 897 is enough to show that 897 is not prime. As soon as you find a factor, you can stop. Begin with divisibility tests –

is 897 even? No, so 2 is not a factor Is the sum of the digits divisible by 3? 8 + 9 + 7 = 24, and 24 is divisible by 3, so we know 897 is divisible by 3.

Therefore 897 is not prime – it is equal to 3 • 299.

Example: Is 887 prime?

We found a factor of 897 fairly quickly, but it’s not so fast here. We must check all the way up to 29, since 30 is the first number whose square is larger than 887 (302 = 900).

is 887 divisible by 2? No, it’s not an even number, so 2 is not a factor, nor is any even number. On the table we can now eliminate checking 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, and 28, because they are multiples of 2.

Is 887 divisible by 3? 8 + 8 + 7 = 23, so 887 is not divisible by 3 or any multiple of 3.

Is 887 divisible by 5? No, 887 does not have 0 or 5 as the last digit. It is not divisible by 5 or any multiple of 5.

Is 887 divisible by 7? No… etc. We find that 887 is prime.

Prime and Composite

http://primes.utm.edu/notes/proofs/infinite/euclids.html

http://www.centreforthemind.com/publications/integerarithmetic.cfm

http://www.youtube.com/watch?v=TmSYXbgcozY

http://www.seti.org/Page.aspx?pid=235

http://en.wikipedia.org/wiki/RSA



The process of eliminating multiples to cut down on the factors to check has a great side effect – if you look at the table generated for 887, you can see that by eliminating multiples, the numbers that are left are primes. The non-highlighted numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29 are all prime numbers. (If you’re a skeptic, check by making factor charts for these numbers.) An ancient and still useful method for finding primes uses this technique. The Sieve of Eratosthenes The Sieve of Eratosthenes is a procedure for finding all the primes less than a given number. Eratosthenes was born in Libya, lived in Alexandria, Egypt as head of the great library and university there in the third century B.C. He has many amazing accomplishments and discoveries, one of which is this method for finding primes. A sieve is a tool used to separate materials by size. Eratosthenes’ sieve separates the numbers into prime and composite. Start with a list or chart of numbers. The list can go as high as you like – 100 is a manageable number.

Remember that 1 is not classified as either prime or composite, so begin by crossing off 1. The next number is 2, and 2 is prime, since the only factor pair that makes 2 is 1 • 2. Circle 2 since it’s prime, but cross off any multiple of 2. These are not prime, since each has 2 as a factor. They fall through the sieve since they are composite numbers.

The next number not crossed out is 3, and 3 is prime. Circle 3, and cross out the multiples of 3.

Real-Life Sieve

A sieve is a common kitchen and industrial tool. The mesh allows small particles to fall through, but keeps larger particles back. The Sieve of Eratosthenes is a metaphorical sieve, allowing composite numbers to

“fall through” and keeping prime numbers back.

http://en.wikipedia.org/wiki/Eratosthenes

Free Pre-Algebra Lesson 9 ! page 3


Lather, rinse, repeat. That is, continue the same way – circle the next number not yet crossed out, then cross out all its multiples. Since 100 = 102, 10 would be the largest number we need to use – but 10 is already crossed out, as are 8 and 9. So it looks like 7 is the last number to check for this chart that goes through 100. The completed sieve for 100 looks like this: Numbers that are composite have been crossed out. The only numbers left are prime. Instead of checking whether or not each number is prime separately, we now have a list of primes under 100. To truly understand this, do one of these things:

• Use the blank chart on the previous page and go through the process yourself. (Best choice!)

• Watch a video of a UCLA professor explaining this, with animation.

• Try this JavaScript application which automates some of the processes. The Prime Factorization of a Number Primes are considered the fundamental multiplicative building blocks of numbers. Every whole number greater than 1 is either prime, or can be written as product of primes (you many need more than two factors, though). The prime factorization of a number expresses the number as a product of primes. To find a prime factorization, start with any factor pair (except the first pair, 1 • number). Here we begin with 12 = 3 • 4. 3 is prime, so circle 3. 4 is not prime, so keep breaking it down – write 4 as the product 2 • 2. 2 is prime, so circle the 2s. 12 is now factored into all primes, and can be written 2 • 2 • 3 = 12. What if you begin with a different factor pair? We could have started with 12 = 2 • 6.

Turns out it makes no difference. 2 is prime, but 6 is not, so keep factoring. 6 = 2 • 3, so in the end, the prime factorization is still 2 • 2 • 3 = 12.

A factoring diagram like this is called a tree diagram. The method works like this:

Example: Find the prime factorization of 70.

Step 1: Find any factor pair of 70. Here we use 7 • 10.

Step 2: Find any factor pair of 10. Here we use 2 • 5.

Step 3: All factors are now prime. Write 70 as the product of primes.

2 • 5 • 7 = 70

7 is prime, 10 is not. Both 2 and 5 are prime. Write the primes in order of size.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

http://www.youtube.com/watch?v=ws5zQDIYccg

http://www.hbmeyer.de/eratosiv.htm



An alternate method.


Step 1: Begin with a list of all primes whose squares are less than the number to be factored.

Step 2: Divide 70 by 2. The result is 35.

Step 3: 2 does not divide 35, so we are done with 2. Cross 2 off the list of primes.

3 does not divide 35, so cross out 3.

5 divides 35, and the result is 7.

Step 4: When the result is prime, you have completed the process. Since 7 is prime, we have complete the prime factorization of 70.

2 3 5 7 2 3 5 7 2 3 5 7 The primes are highlighted.

2 • 5 • 7 = 70

Since the diagrams are “scratch work” to find the factorization, which method you use is entirely up to you.


Scratch Work: Method 1

Method 2

The prime factorization of 60 is 2 • 2 • 3 • 5.


Scratch Work: Method 1

Method 2

The prime factorization of 188 is 2 • 2 • 47.



Uniqueness Every whole number greater than one is either prime or can be written as a product of primes. Either the number has no factors other than itself and 1, and so is prime, or it has some other factors that are prime or can be broken down into primes. But you may wonder if there might be more than one way to write a given number as a product of primes. The answer is no. There is one and only one prime factorization of each composite number. In mathematical language, we say the prime factorization of a number is unique. This is so important it’s called the Fundamental Theorem of Arithmetic. Every whole number greater than one is either prime or can be written as a unique product of primes. In everyday language, things that are unique are one-of-a-kind, individual. And in a way, the prime factorization of a number is a revelation of a kind of number personality, giving you vital information about what kind of number it is, what relatives it has in the number world, and how easy it is to get along with. Measurement systems have frequently chosen subdivisions that have many factors. For example, there are 60 seconds in a minute, and 60 minutes in an hour. Why choose 60? Let’s look at the factor pairs and prime factorization of 60.

The prime factorization of 60 is 60 = 2 • 2 • 3 • 5. It contains the first three prime numbers. Not only that, but each of the first six counting numbers is a factor. This makes 60 a number with a lot of relatives (other numbers with whom it has a factor in common). Numbers like this are easy to divide into parts. If you look at the face of an analog clock, you can see that it’s easy to divide into 12ths (the numbers along the edge), but also into 6ths, 5ths, 4ths, 3rds, etc. We say things like “quarter past” because 15 minutes is 1/4 of 60.

In the Harry Potter series, the wizard monetary system has prime number subdivisions. There are 17 silver sickles to a gold galleon, and 29 bronze knuts to a sickle. The prime numbers give the money a potent “magical” feeling – lucky numbers like 7 and 13 are also primes. The prime factorization of a number is a key to its “personality,” a way to understand the number in a fundamental way. Practically speaking, using the prime factorization makes many fraction operations much simpler.

http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic

Free Pre-Algebra Lesson 9 ! page 6


Lesson 9: Prime Factorization

Worksheet Name

Using the sieve as reference, find the prime factorization of each number. Try both methods explained in the text before making up your mind which you prefer. Write out the completed factorization as a product.

The Completed Sieve for Numbers Through 100.

1. Find the prime factorization of 10.





6. Can you use the prime factorization of 10 to make finding the prime factorization of 100 easier?





1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100




Homework 9A Name

1. Estimate the number of sandbags on the forklift.

2. Find the area and perimeter of a rectangle with length 24 inches and width half of the length.

L = 24, W = ______

A =

P =

3. Evaluate

a. 75/3 + 85/5

b. 75 – 3 • 8

c. 4 • 53

d. 22 + 23 + 24 + 25

e. (3 + 5)3 – 3 + 53

f. 2 • 3 • 5 • 7

g. 5 + 2(7 – 2) + 62 – 4 • 3

h. (45 – 34)2 + (20 – 14)2

4. Use the d = rt formula.

a. A plane travels 540 miles per hour for 6 hours. How far did the plane travel?

b. Sound travels 1,125 feet per second. If a sound has been traveling for 6 seconds, how far has it traveled?

c. IPads were selling at the rate of 200,000 per week. If this rate continued, how many would be sold in 52 weeks (one year)?

d. A 45 rpm record makes 45 revolutions per minute. How many revolutions does the record make in a 3 minute song?

e. Audio CD players read their discs at a 150 kilobytes per second. How many kilobytes are read in a 3,600 second (60 minute) CD?



5. Measure each line to the nearest inch and nearest sixteenth.

a.

b.

c. Draw a line 1 and 5/16 inches long.

6. Find all the pairs of factors of each number.

a. 45 b. 46

c. 47 d. 48

e. 49 f. 50

7. Find the prime factorization. If the number is prime, write “prime.”

a. 25

b. 26 c. 27 d. 28

e. 50 f. 52 g. 54 h. 56




Homework 9A Name

1. Estimate the number of sandbags on the forklift.

3 • 10 • 3 = 90 about 90 sandbags

2. Find the area and perimeter of a rectangle with length 24 inches and width half of the length.

L = 24, W = 12

A = (24)(12) = 288

The area is 288 in2.

P = 2(24 + 12) = 2(36) = 72

The perimeter is 72 inches.

3. Evaluate

a. 75/3 + 85/5

25 + 17 = 42

b. 75 – 3 • 8

75 – 24 = 51

c. 4 • 53

4 • 125 = 500

d. 22 + 23 + 24 + 25

4 + 8 + 16 + 32 = 60

e. (3 + 5)3 – 3 + 53

83 – 3 + 125 = 512 – 3 + 125 = 634

f. 2 • 3 • 5 • 7

210

g. 5 + 2(7 – 2) + 62 – 4 • 3

5 + 2(5) + 36 – 12 = 39

h. (45 – 34)2 + (20 – 14)2

(11)2 + (6)2 = 121 + 36 = 157


a. A plane travels 540 miles per hour for 6 hours. How far did the plane travel?

d = rt = (540)(6) = 3240

The plane has traveled 3,240 miles.

b. Sound travels 1,125 feet per second. If a sound has been traveling for 6 seconds, how far has it traveled?

d = rt = (1125)(6) = 6750

The sound has traveled 6750 feet.

c. iPads were selling at the rate of 200,000 per week. If this rate continued, how many would be sold in 52 weeks (one year)?

d = rt = (200000)(52) = 10400000

10,400,000 iPads sold.

d. A 45 rpm record makes 45 revolutions per minute. How many revolutions does the record make in a 3 minute song?

d = rt = (45)(3) = 135

135 revolutions

e. Audio CD players read discs at a 150 kilobytes per second. How many kilobytes are read in a 3,600 second (60 minute) CD?

d = rt = (150)(3600) = 540000

540,000 kilobytes




The line is 1 and 13/16 inches.

a.

The line is 3/8 inch.

b.




a. 25

25 = 5 • 5

b. 26

26 = 2 • 13

c. 27

27 = 3 • 3 • 3

d. 28

28 = 2 • 2 • 7

e. 50

50 = 2 • 5 • 5

f. 52

52 = 2 • 2 • 13

g. 54

54 = 2 • 3 • 3 • 3

h. 56

56 = 2 • 2 • 2 • 7




Homework 9B Name

1. Estimate the number of bannerfish. (Photo credit)

2. Find the area and perimeter of a rectangle with length 13 cm and width 3 less than the length.

L = 13, W = ______

A =

P =

3. Evaluate

a. 5(15000 – 7000) + 8(7000)

b. (23 – 18)2 + (36 – 23)2

c. 232 – 182 + 362 – 232

d. 9(45)/5 + 32

e. 87 + 72 – 87

f. 17(3)/17


a. A helicopter travels 2 hours at 180 mph. How far does the helicopter travel?

b. Light travels 299,792,458 meters per second. If a laser beam has been traveling for 3 seconds, how far has it traveled?

c. Amazon sells about 150,000 books per day. How many books does Amazon sell in a year (365 days)?

d. About 255 babies are born every minute worldwide. How many babies are born each hour (60 minutes)?

e. Airlines use about 47 plastic cups per second. How many plastic cups do airlines use in one day (86,400 seconds)?

http://commons.wikimedia.org/wiki/File:Jon_hanson_-_schooling_bannerfish_school_(by-sa).jpg




a.

b.



a. 67 b. 68

c. 69 d. 70

e. 71 f. 72


a. 18

b. 19 c. 20 d. 21

e. 36 f. 38 g. 40 h. 42

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Lesson 9 Prime Factorization - Welcome to the Home Page ...voyager.dvc.edu/~lmonth/PreAlg/lesson9student.pdf · Lesson 9 Prime Factorization Some whole numbers, such as 31, have only