2.2 FINDING PRIME FACTORS

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1 2.2 FINDING PRIME FACTORS 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 difficult 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 final problem had her somewhat stumped. What are the prime factors of 250? 2 x 5 x 5 x 5 or 2 x 5 3 See if you can do it. Assess your readiness to complete this activity. Rate how well you understand: Not ready Almost ready Bring it 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 61

2 Chapter 2 Fractions While Example 1 is worked out, step by step, you are welcome to complete Example 2 as a running problem. 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. Special Case: The given number is a prime number (see Model 2) Recall the divisibility test for 2: 504 is even, so it is divisible by = = = 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 = = 7 62

3 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 final division produces one (1) as a quotient is prime = 1 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 Step 6 Present the answer. Present your answer. 504 = or, written in exponent form, 504 = Step 7 Validate your answer. Verify that all factors are prime. Validate that the prime factors are correct by finding their product. 2, 3, and 7 are all prime = = 72 7 =

4 Chapter 2 Fractions Model 1 Determine the prime factorization of the following numbers. Use the methodology. Prime factorization of 630. Prime factorization of Steps 1-4 Steps 1-4 Divisibility test results: even Divisibility test results: not even, but divisible by divisible by divisible by divisible by not divisible by 3 but divisible by divisible by not divisible by 5 but divisible by is prime is prime 1 1 Steps 5 & 6 Steps 5 & = = = = Step 7 Validate: Step 7 Validate: 2, 3, 5, and 7 are all prime. 3, 5, 7, and 11 are all prime = = = = = 90 7 = = 630 = 3465 Model 2 Special Case: The Given Number is a Prime Number Determine the prime factorization of Divisibility test results: not even not divisible by 3 not divisible by 5 If 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. 64

5 Activity 2.2 Finding Prime Factors Try the next prime number: Try 7 Try 11 Try ) ) ) does not divide evenly into does not divide evenly into =49 and 49< =121 and 121< does not divide evenly into 167 Because =169, which is greater than 167, you can stop trying larger primes. Answer: 167 is a prime number. Make Your Own Model 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. Problem: 65

6 Chapter 2 Fractions 1. What is the first prime number and why is it prime? By definition a prime number has exactly two factors, one and the number itself.. The first prime number is What are some methodologies for determining a prime factorization? 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. 3. How do you make sure the prime factors of a number are truly correct? Each factor must be tested to make sure it is prime, then multiply all the factors to get the original number. 4. What are divisibility tests and how do you apply them when finding prime number factors? 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 finding the factors quicker and easier. 5. At what point can you stop applying divisibility tests and conclude that a number is prime? When the factor you are testing is squared and its product is larger than the number to be factored. 6. Skim through the methodologies in the remaining activities of Chapter 2. In which ones do you see prime factorization used? 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 7. What aspect of the model you created is the most difficult to explain to someone else? Explain why. Answers will vary. 66

7 Activity 2.2 Finding Prime Factors Determine the prime factorization of: Validation 1) 48 2) 135 3) 187 4)

8 Chapter 2 Fractions In the second column, identify the error(s) you find 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 Solution What is Wrong Here? Identify Errors or Validate 1) 36 A prime number has only two factors, 1 and itself. 9 is not prime. It is divisible by 3. 2) 105 Divided incorrectly. Correct Process Answer: 36 = 2 x 2 x 3 x 3 or 2 2 x 3 2 Validation 2 x 2 x 3 x 3 = 4 x 9 = 36 3) 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. 68