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1 12B 1.2 Prime Numbers and Factors Factors Terminology: 3 4 =12 The number’s 3 and 4 are called Factors. The result, 12, is called the product. We say that 3 and 4 are factors of 12. However, the number 12 has other factors: 2 6 = 12 1 12= 12 So the complete list of factors for 12 is 1, 2, 3, 4, 6, and 12. Another way of stating what factor are is: “Numbers that divide evenly into a given number”. 1, 2, 3, 4, 6, and 12 all divide evenly into 12. The number 5 is not a factor because it does not divide evenly into 12 (there is a remainder). Example 2. List all the factors of 24. Check sequentially the numbers 1, 2, 3, and so on, to see if we can form any factorizations 24 2 12 3 8 4 6 Answer: The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Your Turn Problem #2 List all the factors of 48. Answer: The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.

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2 12B 1.2 Prime Numbers and Factors A prime number is a whole number greater than 1 that is only divisible by itself and 1. A prime number therefore has only two factors—itself and 1. The Prime Numbers Are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53,... Note: Each of these numbers is only divisible by itself and 1. Note: Because a prime number must be greater than 1, this makes the first prime number, by definition 2 Prime Numbers A composite numbers is a number that is divisible by other numbers besides itself and 1. A composite number has more than two factors. The Composite Numbers Are: (all whole numbers that are not prime): 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25,... Composite Numbers It is important to be able to determine whether a number is prime of composite because it will be a necessary part of the prime factorization process. The prime factorization process will be necessary for concepts such as reducing, adding and subtracting fractions. Determining Prime Numbers Next Slide

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3 12B 1.2 Prime Numbers and Factors Procedure: To determine if a number if a number is prime of composite: Step 1:Use the tests of divisibility to determine if it is divisible by 2, 3, or 5. If none of these work, Step 2:Try to divide the number by the prime numbers beginning with 7, 11, 13, etc. Step 3:Continue this process until the prime number you are dividing is such that when it is multiplied by itself, the product is larger than the number being tested. To summarize: In using the list of prime numbers, if you cannot find one that is divisible into the number being tested, then the number is prime. If you can find a prime number or any other number that is divisible into the number, then the number is composite. Example 3: Determine whether the following are prime or composite: a) 87, b) 97, and c) 539 Solution: a) 87 is composite because it is divisible by 3 (add its digits). b) 97 is not divisible by 2, 3, or 5 (divisibility tests do not work), & it is not divisible by 7 (when you divide it gives a remainder). The next prime number, 11, is not tested because when multiplied by itself, the product, 121, is larger than 97. Answers: 87: composite, 97: prime, 539: composite c) 539 is not divisible by 2, 3, or 5 (divisibility tests do not work), but when we divide by the next prime, 7, we get a quotient with no remainder. Answers: a) composite b) composite c) prime Your Turn Problem #3 Determine whether the following are prime or composite: a) 91, b) 57, and c) 103

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4 12B 1.2 Prime Numbers and Factors Prime factorization is the process of rewriting a composite number as a product of primes. Prime factorization can only be performed on composite numbers. Prime factorization Note: The product of prime factorization must equal the composite number. Determining the Prime Factorization of a Composite Number There are a couple of different methods for obtaining the Prime Factorization. Either method is fine. Tree Method: Example 4: 40 Find two factors whose product is the original number. Draw two branches under the original number and write each factor under each branch. Continue the process until the numbers at below each branch is a prime number Note: Had we begun with two different factors, the result would be the same Your Turn Problem #4 Find the prime factorization of 60

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5 12B 1.2 Prime Numbers and Factors Division Method: Example 5: 315 Your Turn Problem #5 Find the prime factorization of 180 Step 1. Like the process for determining if a number is prime or composite, using the list of primes beginning with "2", determine a prime number that is divisible into the composite. Once that prime number is found, do the division. 315 is not divisible by 2. Move on to 3. Step 2. With the quotient obtained from the previous step, determine if it can, in turn, be divided by that same prime number. Step 3. Continue working with the quotient obtained from each previous step building up from the previous division problem. If the previously used prime number will no longer work, try the next higher prime number. Step 4. Continue process until a prime number is obtained at the top. Step 5. The prime factorization is a listing of the divisors on the left plus the prime number at the top. Then divide by 3 again. Once the prime number no longer works, try the next higher prime number. Since the number on top is prime, we’re done.

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6 12B 1.2 Prime Numbers and Factors The greatest common factor of two or more numbers is the largest number that each of the numbers is divisible by. For example: Find the GCF of 30 and 45. Solution: Both numbers are divisible by 3, 5, and 15. The greatest of those numbers is 15. Answer: 15 is the GCF of 30 and 45. Greatest Common Factor (GCF) Example 6. Find the gcf of 12 and 18. Procedure for finding the GCF Step 1. Find the prime factorization of each number. Step 2. Circle any prime number that appears in all prime factorization lists. It is possible that two or more of any prime number can be common to all lists. Step 3. Multiply the common primes together. If there are no primes in common, write, No GCF. If there is only one prime factor common, then there is nothing to multiply and it becomes the answer. Step 1. Find the prime factorization of each number. Step 2. Circle any prime that appears in both lists. Step 3. Multiply the common primes together. Your Turn Problem #6 Find the gcf of 28 and 42.

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7 12B 1.2 Prime Numbers and Factors Example 7. Find the gcf of 54, 90 and 108. Step 1. Find the prime factorization of each number. Step 2. Circle any prime that appears in both lists. Step 3. Multiply the common primes together. Your Turn Problem #7 Find the gcf of 88, 132 and 220. Then End. B.R

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