7.1 Factors and Greatest Common Factors (GCF) CORD Math Mrs. Spitz Fall 2006

Objectives Find the prime factorization of an integer, and Find the greatest common factor (GCF) for a set of monomials.

Assignment pp #17-22, 29-32, (every 3 rd problem) and 71.

Connection In mathematics, there are many situations where there is more than one correct answer. Suppose three students are asked to draw a rectangle that has an area of 18 square inches. As shown, each student can draw a different rectangle, and each rectangle is correct. 2 in. 9 in. 18 in. 1 in. 3 in. 6 in.

Factors/prime numbers and composite numbers Since 2 x 9, 3 x 6, and 1 x 18 all equal 18, each rectangle has an area of 18 square inches. When two or more numbers are multiplied, each number is a factor of the product. In the example given, 18 is expressed as the product of different pairs of whole numbers. 18 = 2 · 9 18 = 3 · 6 18 = 1 · 18 The whole numbers 1, 18, 2, 9, 3, and 6 are factors of 18. Some whole numbers have exactly two factors, the number itself and1. These numbers are called prime numbers. Whole numbers that have more than two factors are called composite numbers.

Definition of Prime and Composite Numbers A prime number is a whole number, greater than 1, whose only factors are 1 and itself. A composite number is a whole number, greater than 1, that is not prime.

0 and 1 are neither prime nor composite The number 9 is a factor of 18, but not a prime factor of 18, since 9 is not a prime number. When a whole number is expressed as a product of factors that are all prime, the expression is called prime factorization of the number. Thus the prime factorization of 18 is 2 · 3 · 3 or 2 · 3 2.

What else? The prime factorization of every number is unique except for the order in which the factors are written. For example, 3 · 2 · 3 is also a prime factorization of 18, but it is the same as 2 · 3 · 3. This property of numbers is called the Unique Factorization Theorem.

Ex. 1: Find the prime factorization of 84. You can begin by dividing 84 by its least prime factor. Continue dividing by least prime factors until all the factors are prime. The least prime factor of 84 is 2. The least prime factor of 42 is 2. The least prime factor of 21 is 3. All of the factors in the last row are prime. Thus, the prime factorization of 84 is 2 · 2 · 3 · 7 or 2 2 · 3 · 7

Ex. 2: Factor To factor a negative integer, first express it as the product of a whole number and -1. Then find the prime factorization. The least prime factor of 525 is 3. The least prime factor of 175 is 5. The least prime factor of 35 is 5. All of the factors in the last row are prime. Thus, the prime factorization of -525 is -1 · 3 · 5 · 5 · 7 or -1 · 3 · 5 2 · 7 Take the integer and take out the -1.

Ex. 3: Factor 20a 2 b. A monomial is written in factored form when it is expressed as the product of prime numbers and variables where no variable has an exponent greater than 1. The least prime factor of 10 is 2. All of the factors in the last row are prime. Thus, the prime factorization of 20a 2 b is 2 · 2 · 5 · a · a · b or 2 2 · 5 · a · a · b The least prime factor of 20 is 2.

Two or more numbers may have some common factors. Consider the prime factorizations of 90 and 105 shown below. The integers 90 and 105 have 3 and 5 as common prime factors. The product of these prime factors, 3 · 5 or 15, is called the greatest common factor (GCF) of 90 and 105. DEFINITION OF GREATEST COMMON FACTOR: The greatest common factor of two or more integers is the product of the prime factors common to the integers. Note: The GCF of two or more monomials is the product of their common factors, when each monomial is expressed as a product of prime factors.

Ex. 4: Find the GCF of 54, 63, and 180 Factor each number. Then circle the common factors. The GCF of 54, 63, and 180 is 3 · 3 or 3 2 or 9.

Ex. 5: Find the GCF of 8a 2 b and 18a 2 b 2 c Factor each number. Then circle the common factors. The GCF of 8a 2 b and 18a 2 b 2 c is 2 · a · a · b or 2a 2 b.