Chapter 4 Number Theory Copyright 2013 2010 and

4 -2 Prime and Composite Numbers Prime Factorization Number of Divisors Determining if a

Prime and Composite Numbers The following rectangles represent the number 18. 1 18 6 9 2 3 The number 18 has 6 positive divisors: 1, 2, 3, 6, 9 and 18. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Number of Positive Divisors Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Prime Numbers Number of Positive Divisors These numbers have exactly 2 positive divisors, 1

Composite Numbers Number of Positive Divisors These numbers have at least one factor other

Prime and Composite Numbers Number of Positive Divisors The number 1 has only one

Definition Prime number Any positive integer with exactly two distinct, positive divisors Composite number Any whole number greater than 1 that has a positive factor other than 1 and itself Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4 -9 Show that the following numbers are composite. a. 1564 Since 2 | 4, 1564 is divisible by 2 and is composite. b. 2781 Since 3 | (2 + 7 + 8 + 1), 2781 is divisible by 3 and is composite. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4 -9 (continued) c. 1001 Since 11 | [(1 + 0) − (0 + 1)], 1001 is divisible by 11 and is composite. d. 3 · 5 · 7 · 11 · 13 + 1 The product of odd numbers is odd, so 3 · 5 · 7 · 11 · 13 is odd. When 1 is added to an odd number, the sum is even. All even numbers are divisible by 2 and all even numbers, except 2, are composite. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. NCTM grade 7 Curriculum Focal Points, p. 19 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Prime Factorization 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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Prime Factorization 84 495 4 2 21 2 3 5 7 99 11 9

Prime Factorization The two trees produce the same prime factorization, except for the order

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Prime Factorization Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. 1, 2, 3, 4, 6, 8, 12, 24 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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Number of Divisors 1, 2, 3, 4, 6, 8, 12, 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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Number of Divisors Another way to think of the number of positive divisors of 24 is to consider the prime factorization 23 = 8 has four divisors. 3 has two divisors. Using the Fundamental Counting Principle, there are 4 × 2 = 8 divisors of 24. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Number of Divisors If p and q are different primes, m and n are whole number then pnqm has (n + 1)(m + 1) positive divisors. In general, if p 1, p 2, …, pk are primes, and n 1, n 2, …, nk are whole numbers, then has positive divisors. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4 -10 a Find the number of positive divisors of 1, 000. The prime factorization of 1, 000 is 26 has 6 + 1 = 7 divisors, and 56 has 6 + 1 = 7 divisors. has (7)(7) = 49 divisors. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4 -10 b Find the number of positive divisors of 21010. The prime factorization of 21010 is 210 has 10 + 1 = 11 divisors, 310 has 10 + 1 = 11 divisors, 510 has 10 + 1 = 11 divisors, and 710 has 10 + 1 = 11 divisors. has 114 = 14, 641 divisors. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Determining if a Number is Prime Assume that p is a prime greater than 7 and p | 97. Then 97/p also divides 97. Because p ≥ 11, then 97/p must be less than 10 and hence cannot divide 97. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Determining if a Number is Prime 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 a whole number 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 check if any prime less than or equal to is a divisor of n. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. it is enough to

Example 4 -11 a Is 397 composite or prime? The possible primes p such that p 2 ≤ 397 are 2, 3, 5, 7, 11, 13, 17, and 19. Because none of the primes 2, 3, 5, 7, 11, 13, 17, and 19 divide 397, 397 is prime. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4 -11 b Is 91 composite or prime? The possible primes p such that p 2 ≤ 91 are 2, 3, 5, and 7. Because 91 is divisible by 7, it is composite. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.