216 CHAPTER 5 Number Theory and the Real Number System No other natural number listed as a product of primes will have the same prime factorization as 18. The fundamental theorem of arithmetic states this concept formally. (A theorem is a statement or proposition that can be proven true.) The Fundamental Theorem of Arithmetic Every composite number can be expressed as a unique product of prime numbers. In writing the prime factorization of a number, the order of the factors does not matter. However, for consistency, we will write the prime factorization with the factors from smallest to largest and we will use exponents to represent repeated factors. For example, we will write the prime factorization of 18 as 2 3 .2 ⋅ We will illustrate two methods to determine the prime factorization of a number: branching and division. To determine the prime factorization of a number by branching, write the number as the product of two factors. If one or both of the factors are not prime numbers, continue factoring each composite number until all the factors are prime. We illustrate branching in Example 2. Example 2 Prime Factorization by Branching Write 1500 as a product of primes. Solution Select any two numbers whose product is 1500. Among the many choices, two possibilities are 15 100 ⋅ and 30 50. ⋅ First consider 15 100. ⋅ Since neither 15 nor 100 is a prime number, determine any two numbers whose product is 15 and any two numbers whose product is 100. Continue branching as shown in Fig. 5.2 until the numbers in the last row are all prime numbers. To determine the answer, write the product of all the prime factors. The branching diagram is sometimes called a factor tree. Now try Exercise 35 1500 100 15 3 3 5 10 10 2 5 5 2 5 Figure 5.2 1500 50 30 6 5 5 10 2 5 2 3 5 5 Figure 5.3 We see that the numbers in the last row of factors in Fig. 5.2 are all prime numbers. Thus, the prime factorization of 1500 is ⋅ ⋅ ⋅ ⋅ ⋅ = 3 5 2 5 2 5 ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ 223555 2 35. 2 3 Note from Fig. 5.3 that choosing 30 and 50 as the first pair of factors also leads to the same prime factorization of 2 3 5 . 2 3 ⋅ ⋅ 7 To obtain the prime factorization of a number by the division method, divide the given number by the smallest prime number by which it is divisible. Place the quotient under the given number. Then divide the quotient by the smallest prime number by which it is divisible and again record the quotient. Repeat this process until the quotient is a prime number. The prime factorization is the product of all the prime divisors and the prime (or last) quotient. This procedure is illustrated in Example 3.
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