Objectives Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Prime Numbers and GCF 1. Determine if a number is prime, composite, or neither. 2. Find the prime factorization of a given number. 3. Find all factors of a given number. 4. Find the greatest common factor of a given set of numbers by listing. 5. Find the greatest common factor of a given set of numbers using prime factorization. 6. Find the greatest common factor of a set of monomials. 3.5

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Determine if a number is prime, composite, or neither.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definition Prime number: A natural number that has exactly two different factors, 1 and itself. Is 1 prime? No because a prime must have exactly two different factors and 1’s factors are the same 1 and 1. Is 2 prime? Yes, because 2 is a natural number whose only factors are 1 and 2 (itself). Is 3 prime? Yes, because 3 is a natural number whose only factors are 1 and 3 (itself). Is 4 prime? No, because 4 has factors other than 1 and 4 (2 is also a factor).

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley If we continue this line of thinking, we get the following list of prime numbers… 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, … What about some of the numbers that are not in this list…what are they called?

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definition Composite number: A natural number that has factors other than 1 and itself. Is 1 composite? No, because its only factor is itself. The number 1 is neither prime or composite. The first composite number is 4. Here is a short list of composite numbers. 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25,… Remember…every composite number is divisible by at least one prime number.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley How do we determine if a number is prime or composite when it isn’t immediately and obviously apparent? Here’s an example… Is 91 prime or composite? Is 91 divisible by 2? Is 91 divisible by 3? Is 91 divisible by 5? Is 91 divisible by 7?

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Is 127 prime or composite? Divide by each of the prime numbers… Divisible by 2? Divisible by 3? Divisible by 5? Divisible by 7? What about 11?

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To determine if a given number is prime or composite, divide it by the prime numbers in ascending order and consider the results. Procedure 1.If the given number is divisible by the prime, stop. The given number is a composite number. 2.If the given number is not divisible by the prime, consider the quotient. a.If the quotient is greater than the current divisor, repeat the process with the next prime on the list of prime numbers. b.If the quotient is equal to or greater than the current prime divisor, stop. The given number is prime.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 Determine if 157 is prime or composite. Solution: Divide by the list of prime numbers. Is 157 divisible by 2? Is 157 divisible by 3? Is 157 divisible by 5? Is 157 divisible by 7? s 157 divisible by 11? s 157 divisible by 13?

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 2 Find the prime factorization of a given number.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definition Prime factorization: A product written with prime factors only. For example, the prime factorization of 20 is 2 2 5, which we can write in exponential form as or

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To find the prime factorization of a composite number using a factor tree: Procedure 1.Draw two branches below the number. 2.Find two factors whose product is the given number and place them at the end of the two branches. 3.Repeat steps 1 and 2 for every composite factor. 4.Write a multiplication sentence containing all the prime factors.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 2 We could have started a factor tree for 84 many different ways and still ended up with the same result. 84

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 2 Find the prime factorization. Write the answer in exponential form. b. 96c. 3500

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 3 Find all factors of a given number.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3 List all factors of 24. List all factors of 45. List all factors of 60.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 4 Find the greatest common factor of a given set of numbers by listing.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definition Greatest common factor: The greatest number that divides all given numbers with no remainder. The greatest common factor is sometimes called the greatest common divisor.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To find the greatest common factor by listing: Procedure 1.List all factors for each given number. 2.Search the lists for the greatest factor common to all lists.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 4 Find the GCF of 24 and 60.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 4 Find the GCF of 48 and 72.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 5 Find the greatest common factor of a given set of numbers by using prime factorization.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To find the greatest common factor using prime factorization: Procedure 1.Write the prime factorization of each number in exponential form. 2.Create a factorization for the GCF that contains only those prime factors common to all factorizations, each raised to the least of its exponents. 3.Multiply. Note: If there are no common prime factors, the GCF is 1.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 5 Find the GCF of 3024 and 2520.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 The floor of a 42 foot by 30 foot room will be covered with colored squares like a checkerboard. All the squares must be equal sized with whole number dimensions and may not be cut or overlapped to fit inside the room. Find the dimensions of the largest possible square that can be used.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 6 Find the greatest common factor of a set of monomials

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 7 Find the GCF of the monomials. a. 18x 4 and 12x 3 b. 24n 2 and 40n 5 c. 14t 4 and 45