61

2.2 FINDING PRIME FACTORS

Assess your readiness to complete this activity. Rate how well you understand: Not ready

Almost ready

Bringit on!

• the characteristics of prime numbers and composite numbers

• the terminology and notation associated with prime factorization

• how to determine the prime factorization of a number

• how to validate that the prime factorization of a number is accurate

• Correctly identifying a number as prime or composite

• Writing any composite number as a product of its prime factors– following a methodology for determining the prime factorization– validation of the prime factorization

Lauren was talking to her uncle, a computer programmer, about her math class. She said that while she thought prime numbers were interesting, she just didn’t see much point to them, especially in the world outside of class. Her uncle smiled and asked Lauren if she ever purchased anything online.

“Of course,” she said. “Doesn’t almost everybody?”

He replied, “Yeah, millions of people do. And every time you buy something online, you should stop and thank prime numbers for keeping your credit card or bank information secure. Modern data encryption relies on the product of two large prime numbers being very diffi cult and time-consuming to factor. Those prime numbers are used to code and decode your information, keeping it safe.”

With a new appreciation for the importance and utility of prime numbers, Lauren worked to complete her homework. The fi nal problem had her somewhat stumped.

See if you can do it.

What are the prime factors of 250?

2 x 5 x 5 x 5 or 2 x 53

62

Chapter 2 — Fractions

Example 1: Determine the prime factorization of 504.

Example 2: Determine the prime factorization of 90 using this methodology. Try It!

Steps in the Methodology Example 1 Example 2

Step 1

Write the number.

Set up the number with enough work space under it for divisions.

504

Step 2

Divide by its smallest prime number factor.

Divide the number by the smallest prime number that will divide into it evenly, and write your result below it.

Continue to divide by that same prime number until it is no longer a factor of your result.

The given number is a prime number (see Model 2)

Special Case:

Recall the divisibility test for 2:

504 is even, so it is divisible by 2.

2 504

2 252

2 126

63

504 ÷ 2 = 252

252 ÷ 2 = 126

126 ÷ 2 = 63

Step 3

Divide by the next prime number factor.

Repeat Step 2, dividing by the next larger prime number that is a factor of your result.

Continue to perform divisions until that prime number no longer divides evenly.

Recall the divisibility test for 3:

If 3 divides the sum of the digits, it divides the number.

2 504

2 252

2 126

3 63

3 21

7

63 ÷ 3 = 21

21 ÷ 3 = 7

While Example 1 is worked out, step by step, you are welcome to complete Example 2 as a running problem.

63

Activity 2.2 — Finding Prime Factors

Steps in the Methodology Example 1 Example 2

Step 4

Divide by prime numbers until the quotient is one (1).

Continue dividing by the next larger prime that is a factor of the result until the fi nal division produces one (1) as a quotient.

2 504

2 252

2 126

3 63

3 21

7 7

1 7 ÷ 7 = 1

7 is prime

The prime factoring is complete.

Step 5

Collect prime divisors.

Collect all of the prime factors on the left side, from smallest to largest, and use each as many times as it appears.

2×2×2×3×3×7

Step 6

Present the answer.

Present your answer. 504 = 2×2×2×3×3×7or, written in exponent form,

504 = 23 × 32 × 7

Step 7

Validate your answer.

Verify that all factors are prime.

Validate that the prime factors are correct by fi nding their product.

2, 3, and 7 are all prime

2×2×2×3×3×7 =8 × 9 × 7 =

72 × 7 = 504

64

Chapter 2 — Fractions

Model 1

Model 2

Determine the prime factorization of the following numbers. Use the methodology.

Prime factorization of 630. Prime factorization of 3465.

Steps 1-4 Steps 1-4

Divisibility test results: Divisibility test results:

Steps 5 & 6 Steps 5 & 6

630 = 2 × 3 × 3 × 5 × 7 3465 = 3 × 3 × 5 × 7 × 11

= 2 × 32 × 5 × 7 = 3

2 × 5 × 7 × 11

Step 7 Validate: Step 7 Validate:

2, 3, 5, and 7 are all prime. 3, 5, 7, and 11 are all prime.

2 × 3 × 3 × 5 × 7 = 2 × 9 × 5 × 7 3 × 3 × 5 × 7 × 11 = 9 × 5 × 7 × 11

= 18 × 5 × 7 = 45 × 7 × 11

= 90 × 7 = 315 × 11

= 630 = 3465

2 630

3 315

3 105

5 35

7 7

1

even

divisible by 3

divisible by 3

divisible by 5

7 is prime

3 3465

3 1155

5 385

7 77

11 11

1

not even, but divisible by 3

divisible by 3

not divisible by 3 but divisible by 5

not divisible by 5 but divisible by 7

11 is prime

Special Case: The Given Number is a Prime Number

Determine the prime factorization of 167.

Divisibility test results:

167If you are uncertain as to whether a number is prime, try in succession the next larger prime numbers. You can stop when the prime number you are testing times itself is greater than the number.

not even

not divisible by 3

not divisible by 5

65

Activity 2.2 — Finding Prime Factors

Try the next prime number:

7×7=49 and 49<167 11×11=121 and 121<167

Because 13 × 13 =169, which is greater than 167, you can stop trying larger primes.

Answer: 167 is a prime number.

Try 7 Try 11 Try 13

)7 1671427216

23

−

−

)11 1671157552

15

−

−

)13 16713372611

12

−

−

7 does not divide evenly into 167

11 does not divide evenly into 167

13 does not divide evenly into 167

Make Your Own Model

Problem: _________________________________________________________________________

Either individually or as a team exercise, create a model demonstrating how to solve the most diffi cult problem you can think of.

Answers will vary.

66

Chapter 2 — Fractions

1. What is the fi rst prime number and why is it prime?

2. What are some methodologies for determining a prime factorization?

3. How do you make sure the prime factors of a number are truly correct?

4. What are divisibility tests and how do you apply them when fi nding prime number factors?

5. At what point can you stop applying divisibility tests and conclude that a number is prime?

6. Skim through the methodologies in the remaining activities of Chapter 2. In which ones do you see prime factorization used?

7. What aspect of the model you created is the most diffi cult to explain to someone else? Explain why.

Prime factor numbers by mental math, breaking down numbers into their factors mentally; factor trees, by branching out factors in a diagram, or the tile method (preferred method), showing successive divisions of prime factors.

By defi nition a prime number has exactly two factors, one and the number itself.. The fi rst prime number is 2.

Reducing Fractions, Multiplying and Dividing Fractions, Determining LCM, Building Equivalent Fractions for a Given Set of Factors, Ordering Fractions, Adding Fractions and Mixed Numbers, Subtracting Fractions and Mixed

Each factor must be tested to make sure it is prime, then multiply all the factors to get the original number.

When the factor you are testing is squared and its product is larger than the number to be factored.

Divisibility tests are shortcut methods to decide if a number is divisible by certain factors, such as 2, 3, and 5. Using the divisibility rules gives you a quick start to determine if one of the lower primes is a factor of the number being represented in the product of primes form. This makes fi nding the factors quicker and easier.

Answers will vary.

67

Activity 2.2 — Finding Prime Factors

Determine the prime factorization of: Validation

1) 48

2) 135

3) 187

4) 127

68

Chapter 2 — Fractions

In the second column, identify the error(s) you fi nd in each of the following worked solutions. If the answer appears to be correct, validate it in the second column and label it “Correct.” If the worked solution is incorrect, solve the problem correctly in the third column and validate your answer in the last column.

The directions are to determine the prime factorization of each of the following numbers.

Worked SolutionWhat is Wrong Here?

Identify Errors or Validate Correct Process Validation

1) 36 A prime number has only two factors, 1 and itself.9 is not prime.It is divisible by 3.

2 36

2 18

3 9

3 3

1

Answer:36 = 2 x 2 x 3 x 3

or 22 x 32

2 x 2 x 3 x 3= 4 x 9= 36

2) 105 Divided incorrectly.

3) 51 51 is not prime. If you add the digits, 51 will be divisible by 3.

4) 99 One (1) is not a prime number.

Do not use it as a factor.