5-4 Prime and Composite Numbers Prime and Composite Numbers Prime Factorization Number of Divisorss Determining if a Number is Prime More About Primes Prime and Composite Numbers Students should recognizee that different types of numbers have particular characteristics; for example, square numbers have an odd number of factors and prime numbers have only two factors. The following rectangles represent the number 18. The number 18 has 6 positive divisors: 1, 2, 3, 6, 9 and 18. Number of Positive Divisors Prime and Composite Numbers Below each number listed across the top, we identify numbers less than or equal to 37 that have that number of positive divisors.
Prime number Any positive integer with exactly two distinct, positive divisors Composite number Any integer greater than 1 that has a positive factor other than 1 and itself Example 1 Show that the following numbers are composite. a. 1564 b. 2781 c. 1001 d. 3 5 7 11 13 + 1 Prime Factorization Students continue to develop their understanding of multiplication and division and the structure of numbers by determining if a counting number greater than 1 is a prime, and if it is not, by factoring it into a product of primes. Composite numbers can be expressed as products of two or more whole numbers greater than 1. Each expression of a number as a product of factors is a factorization. A factorization containing only prime numbers is a prime factorization. Fundamental Theorem of Arithmetic (Unique Factorization Theorem) Each composite number can be written as a product of primes in one and only one way except for the order of the prime factors in the product.
Prime Factorization To find the prime factorization of a composite number, rewrite the number as a product of two smaller natural numbers. If these smaller numbers are both prime, you are finished. If either is not prime, then rewrite it as the product of smaller natural numbers. Continue until all the factors are prime. The two trees produce the same prime factorization, except for the order in which the primes appear in the products. Prime Factorization We can also determine the prime factorization by dividing with the least prime, 2, if possible. If not, we try the next larger prime as a divisor. Once we find a prime that divides the number, we continue by finding smallest prime that divides that quotient, etc.
Number of Divisors How many positive divisors does 24 have? We are not asking how many prime divisors, just the number of divisors any divisors. Since 1 is a divisor of 24, then 24/1 = 24 is a divisor of 24. Since 2 is a divisor of 24, then 24/2 = 12 is a divisor of 24. Since 3 is a divisor of 24, then 24/3 = 8 is a divisor of 24. Since 4 is a divisor of 24, then 24/4 = 6 is a divisor of 24. Another way to think of the number of positive divisors of 24 is to consider the prime factorization 2 3 = 8 has four divisors. 3 has two divisors. Using the Fundamental Counting Principle, there are 4 2 = 8 divisors of 24.
If p and q are different primes, then p n q m has (n + 1)(m + 1) positive divisors. In general, if p 1, p 2,, p k are primes, and n 1, n 2,, n k are whole numbers, then has positive divisors. Example 2 a. Find the number of positive divisors of 1,000,000. b. Find the number of positive divisors of 210 10. Determining if a Number is Prime To determine if a number is prime, you must check only divisibility by prime numbers less than the given number. For example, to determine if 97 is prime, we must try dividing 97 by the prime numbers: 2, 3, 5, and so on as long as the prime is less than 97. If none of these prime numbers divide 97, then 97 is prime. Upon checking, we determine that 2, 3, 5, 7 do not divide 97. Assume that p is a prime greater than 7 and p 97. Then 97/p also divides 97. Because p 11, then 97/pp must be less than 10 and hence cannot divide 97.
If d is a divisor of n, then is also a divisor of n. If n is composite, then n has a prime factor p such that p 2 n. If n is an integer greater than 1 and not divisible by any prime p, such that p 2 n, then n is prime. Note: Because p 2 n implies that than or equal to is a divisor of n. it is enough to check if any prime less Example 3 a. Is 397 composite or prime? b. Is 91 composite or prime? Sieve of Eratosthenes One way to find all the primes less than a given number is to use the Sieve of Eratosthenes. If all the natural numbers greater than 1 are considered (or placed in the sieve), the numbers that are not prime are methodically crossed out (or drop through the holes of the sieve). The remaining numbers are prime. 5.4 Homework # A-2, 3, 4, 5, 6, 7, 9, 10, 13, 14, 18, 19