Factors and the Greatest Common Factor

What numbers can be multiplied together to get 24? The number 24 is called the product because it is the result of multiplying two or more numbers together. 1 x 24 2 x 12 3 x 8 The numbers 1, 2, 3, 4, 6, 8, 12, and 24 are called factors. These are the numbers being multiplied together. 4 x 6 What are the factors of seven? 1 x 7 Composite numbers – are numbers which have two or more factors like the number 24. Prime numbers – are numbers having exactly two factors: one and itself. Seven is a prime number.

The prime factorization of 56 is 2 • 2 • 2 • 7 or 23 • 7. Factor 56 using only prime factors. To do this, we can make a factor tree as follows. 56 28 is composite so we can factor it further. 2 is prime. 2 • 28 14 is composite, so we can factor it further. 2 is prime. 2 • 14 2 is prime. 2 • 7 is prime. 7 The prime factorization of 56 is 2 • 2 • 2 • 7 or 23 • 7.

The prime factorization of 56 is 2 • 2 • 2 • 7 or 23 • 7. We can make a factor tree using different starting factors. 56 14 is composite so we can factor it further. 4 is composite so we can factor it further. 4 • 14 • 2 • 2 2 7 2 is prime. 7 is prime. The prime factorization of 56 is 2 • 2 • 2 • 7 or 23 • 7. The prime factorization of every number is unique. No matter what factors you choose, the prime factorization will still be the same for that number.

The greatest common factor of 60 and 105 is 3 • 5 or 15. Find the prime factorization of the following two numbers. 60 105 2 • 30 5 • 21 2 • 15 3 • 7 60 = 2 • 2 • 3 • 5. 3 • 5 105 = 3 • 5 • 7 What factors do these two numbers have in common? 3 and 5 The greatest common factor of 60 and 105 is 3 • 5 or 15.

You Try It. Find the greatest common factor (GCF) of the given monomials. 1. 34, 85 2. 14p4q, 35pq3 3. 15x2, 30xy2, 57x3y Determine whether each number is prime or composite. If the number is composite, find its prime factorization. 4. 145 5. 113

Problems 1-3 34 = 2 • 17 85 = 5 • 17 The greatest common factor (GCF) of 34 and 85 is 17. 14p4q = 2 • 7 • p • p • p • p • q 35pq3 = 5 •7 • p • q • q • q The GCF is 7pq . 15x2 = 3 • 5 • x • x 30xy2 = 2 • 3 • 5 • x • y • y 57x3y = 3 • 19 • x • x • x • y The GCF is 3x .

Answer for Problems 4 and 5 145 is composite. 145 = 5 • 29 113 is a prime number.