1. Copyright © 2016, 2013, and 2010, Pearson Education, Inc.4
Chapter
Number Theory
Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

2. 4-2 Prime and Composite Numbers
Students will be able to understand and explain
• Prime and Composite numbers.
• The number of divisors of any whole number.
• The Fundamental Theorem of Arithmetic.
• The factorization of whole numbers.

3. 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.

4. Prime and Composite Numbers
Number of Positive Divisors
Below each number listed across the top, we identify numbers less than or equal to 37 that have that number of positive divisors.

5. Prime Numbers Number of Positive Divisors
These numbers have exactly 2 positive divisors, 1 and themselves.

6. Composite Numbers Number of Positive Divisors
These numbers have at least one factor other than 1 and themselves.

7. Prime and Composite Numbers
The number 1 has only one positive factor – it is neither prime nor composite.
Number of Positive Divisors

8. Definition 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

9. Example Show that the following numbers are composite. a. 1564
Since 2 | 4, 1564 is divisible by 2 and is composite.
b. 2781
Since 3 | ( ), 2781 is divisible by 3 and is composite.

10. Example (continued)
c. 1001
Since 11 | [(1 + 0) − (0 + 1)], 1001 is divisible by 11 and is composite.
d. 3 · 5 · 7 · 11 ·
The product of odd numbers is odd, so · 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.

11. 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.

12. 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.

13. 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.

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

15. Prime Factorization
The two trees produce the same prime factorization, except for the order in which the primes appear in the products.

16. 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.

17. Prime Factorization

18. 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.

19. 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.

20. 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.

21. 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 p1, p2, …, pk are primes, and n1, n2, …, nk are whole numbers, then has whole number divisors.

22. Example Find the number of positive divisors of 1,000,000.
The prime factorization of 1,000,000 is
26 has = 7 divisors, and 56 has = 7 divisors.
has (7)(7) = 49 divisors.

23. Example Find the number of positive divisors of 21010.
The prime factorization of is
210 has = 11 divisors, 310 has = 11 divisors, 510 has = 11 divisors, and 710 has = 11 divisors.
has 114 = 14,641 divisors.

24. 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.

25. 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.

26. 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
p2 ≤ n.
If n is an integer greater than 1 and not divisible by any prime p, such that p2 ≤ n, then n is prime.
Note: Because p2 ≤ n implies that it is enough to check if any prime less than or equal to is a divisor of n.

27. Example Is 397 composite or prime?
The possible primes p such that p2 ≤ 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.

28. Example Is 91 composite or prime?
The possible primes p such that p2 ≤ 91 are 2, 3, 5, and 7.
Because 91 is divisible by 7, it is composite.

29. 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.

30. Sieve of Eratosthenes