14 Chapter 1 Real Numbers and Algebraic Expressions ������������������������ A number p is a prime number if its whole number factors are only 1 and the number p itself. A number that is not prime is called composite. This means that the number has factors other than 1 and itself. Examples of composite numbers are 4, 6, and 16, and their factored forms are shown. Number Factored Forms 4 1 · 4 or 2 · 2 6 1 · 6 or 2 · 3 16 1 · 16 or 2 · 8 or 4 · 4 The number 1 is neither prime nor composite. Every number, other than 1, can be expressed as a product of prime factors. This factorization is called the prime factorization of the number. We can find the prime factorization of 60, for example, using a factor tree. We begin with any two factors of 60, such as 6 and 10 or 2 and 30. We find factors of any number that is not prime and continue the process until all factors are prime numbers. When we find a prime factor, we circle it for easy reference, as shown. 60 2 30 2 15 3 5 60 6 10 2 3 2 5 So, the prime factorization of 60 is 2 · 3 · 2 · 5 or 22 · 3 · 5 because the factors 2, 3, and 5 are prime numbers. Note that it doesn’t matter which two factors we choose to begin the factor tree, because we will always obtain the same factorization except for the order of the factors. ���������������������� Writing the Prime Factorization of a Number Step 1: Write the number as a product of any two of its factors. Step 2: If both factors are prime, then the factorization is complete. Step 3: If either of the factors is not prime, rewrite that factor as a product. If these factors are prime, then the factorization is complete. Step 4: Continue this process until all factors are prime. �� ������������������������ ������������������ Write the prime factorization of each number. 1a. 42 1b. 120 Solutions 1a. 42 Write 42 as a product of any two of its factors. Circle the prime factor, 7. 6 7 2 3 Rewrite 6 as 2 · 3. The prime factorization of 42 is 2 · 3 · 7.
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