214 CHAPTER 5 Number Theory and the Real Number System The number 5 is a prime number because it is divisible only by the factors 1 and 5. The first eight prime numbers are 2, 3, 5, 7, 11, 13, 17, and 19. The number 2 is the only even prime number. All other even numbers have at least three divisors: 1, 2, and the number itself. Definition: Prime Number A prime number is a natural number greater than 1 that has exactly two factors (or divisors), itself and 1. Profile in Mathematics Eratosthenes of Cyrene Eratosthenes of Cyrene (275– 195 B.C. ) was born in northern Africa near the present-day city of Shahhat, Libya. Eratosthenes is best known for being the first to estimate accurately the diameter of Earth. He is also credited for developing a method of determining prime numbers known as the sieve of Eratosthenes. Although he is most known for his work in mathematics, Eratosthenes also was influential in the fields of history, geography, and astronomy. In addition, Eratosthenes served for many years as the director of the famous library in Alexandria, Egypt. Although Eratosthenes was a highly regarded scholar throughout the ancient world, only fragments of his writing remain today. Eratosthenes was near 80 years old when, after losing his vision, he died from voluntary starvation. Definition: Composite Number A composite number is a natural number that is divisible by a number other than itself and 1. Any natural number greater than 1 that is not prime is composite. The first eight composite numbers are 4, 6, 8, 9, 10, 12, 14, and 15. The number 1 is neither prime nor composite; it is called a unit . The number 38 has at least three divisors, 1, 2, and 38, and hence is a composite number. In contrast, the number 23 is a prime number, since its only divisors are 1 and 23. More than 2000 years ago, the ancient Greeks developed a technique for determining which numbers are prime numbers and which are not. This technique is known as the sieve of Eratosthenes , for the Greek mathematician Eratosthenes of Cyrene, who first used it. To determine the prime numbers less than or equal to any natural number, say, 50, using this method, list the first 50 counting numbers (Fig. 5.1). Cross out 1, since it is not a prime number. Circle 2, the first prime number. Then cross out all the multiples of 2: 4, 6, 8, , 50. … Circle the next prime number, 3. Cross out all multiples of 3 that are not already crossed out. Continue this process of crossing out multiples of prime numbers until you reach the prime number p, such that p p, ⋅ or p ,2 is greater than the last number listed, in this case 50. Therefore, we next circle 5 and cross out its multiples. Then circle 7 and cross out its multiples. The next prime number is 11, and 11 11, ⋅ or 121, is greater than 50, so you are done. At this point, circle all the remaining numbers to obtain the prime numbers less than or equal to 50. The prime numbers less than or equal to 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47. Figure 5.1 1 2 12 22 32 42 11 21 31 41 3 14 24 34 44 13 23 33 43 5 6 16 26 36 46 15 25 35 45 7 8 18 28 38 48 17 27 37 47 9 19 29 39 49 10 20 30 40 50 4 Rules of Divisibility Now we turn our attention to composite numbers and their factors. The rules of divisibility given in the following chart are helpful in determining divisors (or factors) of composite numbers.
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