Copyright © Cengage Learning. All rights reserved.

CHAPTER 4

Number Theory

Copyright © Cengage Learning. All rights reserved.

SECTION 4.2

Prime and Composite Numbers

3

What Do You Think?

• Why is 1 neither a prime nor a composite number?

• How are prime numbers the building blocks for the set of natural numbers?

4

Prime and Composite Numbers

Over the centuries, many famous mathematicians have been fascinated by prime numbers. The Greeks are believed to have discovered prime numbers and were also fascinated by them.

The prime numbers are the building blocks of all natural numbers greater than 1. Children are often intrigued by prime numbers.

Over the years, I have found that my strongest students are the ones who not only know facts and procedures but also have good number sense.

5

Prime and Composite Numbers

One of the ways to develop this number sense in elementary children is through investigations that deal with prime numbers.

A natural number is a prime number if it has exactly two factors: 1 and itself. A natural number that has more than two factors is called a composite number.

The number 1 doesn’t fit into either of the sets above, and so we say that 1 is neither a prime number nor a composite number.

6

Determining Whether a Number Is Prime or Composite

7

Determining Whether a Number Is Prime or Composite

A question that has fascinated both ancient and modern mathematicians concerns being able to determine whether a large number is a prime number or a composite number. We don’t need a large number for this question to be difficult.

For example, is the number 103 prime or composite?Immediately, we can see that no even number will divide 103.

However, without sophisticated techniques, in order to determine if 103 is prime, we must check to see whether it is divisible by each odd number up to 51.

8

Determining Whether a Number Is Prime or Composite

Thus we would have to check 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, and 33. Do you see why we can stop at 33?

There is a very elegant shortcut that saves us from having to test all these numbers.

This shortcut was discovered over 2000 years ago.

9

Investigation A – The Sieve of Eratosthenes

The sieve of Eratosthenes was developed by the Greek mathematician Eratosthenes, who lived about 230 B.C., as a tool for determining all prime numbers less than a given number.

As usual, you will get far more from the following activity if you do it with pencil and paper rather than just reading it.

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Investigation A – The Sieve of Eratosthenes

In Table 4.2, cross out 1, which is neither composite nor prime.

Circle 2, which is the first prime number. Now cross out all multiples of 2.

Table 4.2

cont’d

11

Investigation A – The Sieve of Eratosthenes

Circle the next unmarked number (3), which must be prime, and cross out all multiples of 3. Did you make use of any patterns to help you cross out multiples of 3?

Circle the next unmarked number (5), and then cross out all multiples of 5.

Circle the next unmarked number (7) and stop. On the basis of the crossing out that you have already done, what do you predict will be the first multiple of 7 that you will have to cross out?

cont’d

12

Investigation A – The Sieve of Eratosthenes

You found that the first multiple of 7 that wasn’t already crossed out was 49—that is 7 7. The other multiples of 7 that you had to cross out were 7 11, 7 13, and 7 17.

Now circle the next prime number (11), and stop. Before you cross out all multiples of 11, what do you predict will be the first multiple of 11 that you will have to cross out?

You found that the first multiple of 11 that wasn’t already crossed out was 121—that is 11 11. Now circle all the numbers in the table up to 121 that have not yet been crossed out. What do you think these numbers have in common?

cont’d

13

If your intuition said that these are all prime numbers, you are right. Can you explain why all these numbers must beprime numbers? For example, can you explain why 149, forexample, cannot be a multiple of 7, 11, 13, 17, 19, or any other prime number less than 149?

One way of demonstrating why all the remaining numbers must be prime involves indirect reasoning, which is a method of logic in which we prove that something is falseby assuming that it is true and then showing that this assumption leads to a contradiction.

Investigation A – Discussion

14

Investigation A – Discussion

Let us assume that one of the circled numbers in the sieve, for example 149, is not prime.

• 149 is clearly not a multiple of 2, because it’s an odd number.

• 149 cannot be a multiple of 3, because it would have been crossed out with the multiples of 3.

• 149 cannot be a multiple of 5, because it would havebeen crossed out with the multiples of 5.

• 149 cannot be a multiple of 7, because it would havebeen crossed out with the multiples of 7.

cont’d

15

Investigation A – Discussion

• 149 cannot be a multiple of 11, because it would have been crossed out with the multiples of 11.

• 149 cannot be a multiple of 13, because we know from our investigation with the sieve that the first possible multiple of 13 that hasn’t already been crossed out would be 13 13, which is 169.

In similar fashion, we can show that 149 cannot be a multiple of any other number.

This disproves the initial assumption that 149 was not prime. Therefore, it is impossible that 149 is a composite number, so it must be a prime number.

cont’d

16

Investigation A – Discussion

If we begin to cross out multiples of 13, we find that all the multiples of 13 from 1 13 to 12 13 have already beencrossed out.

Therefore, the first multiple of 13 that we needed to cross out is 13 13. But 13 13 is 169, which is greater than 149, and so we know that 13 cannot be a factor of 149.

Similarly, all numbers greater than 13 (which must have squares that are greater than 169) cannot be factors of 149. We can turn this observation into a generalization: If the square of a number is greater than the number weare testing for, we can stop.

cont’d

17

Determining Whether a Number Is Prime or Composite

This very powerful generalization, in turn, can be stated as a rule:

Test for determining whether a number n is prime:List all the prime numbers p that satisfy the equation

P or (p2 n).

If none of those prime numbers divides n, then n is a prime number.

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Prime Factorization

19

Prime Factorization

One way to find all the factors of a number is to determine the prime factorization of that number—that is, torepresent that number as the product of numbers, each ofwhich is prime.

The prime factorization of 96 is 2 2 2 2 2 3.

Do you see how this representation can help us to determine all the factors of 96?

We simply check to see whether our number can be constructed from the prime factorization. Clearly, 2 and 3are factors of 96.

20

Prime Factorization

• 4 is a factor because 4 = 2 2.

• 5 is not a factor of 96 because 5 is not in this set.

• 6 is a factor because 6 = 2 3.

• 7 is not a factor of 96 because 7 is not in this set.

• 8 is a factor because 8 = 2 2 2.

• 9 is not a factor of 96 because 3 3 is not in this set.

• 10 is not a factor of 96 because 2 5 is not in this set.

• 11 is not a factor of 96 because 11 is not in this set.

• 12 is a factor because 12 = 2 2 3.

21

Prime Factorization

Because 8 12 = 96, we don’t have to look any further. Do you see why?

In this case, each factor up to 12 is part of a pair: 1 96, 2 48, 3 32, 4 24, 6 16, and 8 12. Thus the factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96.

Determining the prime factorization of a number is not an easy task for many students, and there are many different methods.

22

Prime Factorization

Tree diagrams and prime factorization One technique for determining the prime factorization of a number that appeals to many students is to use a tree diagram. The top of the tree is the original number.

Each number is equal to the product of the two numbers immediately below it. Below are two tree diagrams that produce the prime factorization for 36—that is, 2 2 3 3.

23

Prime Factorization

Division and prime factorization Other students prefer a technique that looks like long division that goes up instead of down. The division is done mentally.

Some students like to do it “in order”; that is, they divide by 2 until they can do it no more and then go to bigger and bigger prime numbers. When they do this, their final answer is already in order from smallest to largest.

24

The Fundamental Theorem of Arithmetic

25

The Fundamental Theorem of Arithmetic

If you compare the two tree diagrams above, you can see that we got the same prime factorization for 36 regardless of how we started. It turns out that this is true in all cases.

This seemingly obvious statement actually has quite a formal name, the Fundamental Theorem of Arithmetic.

It states that every natural number greater than 1 has one unique way of being represented as a product of its prime factors.

26

A Largest Prime Number?

27

A Largest Prime Number?

There is no largest prime number. The first proof of this conjecture occurs in Euclid’s Elements.

The proof uses indirect reasoning, in which we begin by assuming that there is a largest prime number, which we will call P.

Let us now create a number X, in the following way:

X = (2 3 4 ….P) + 1

We have assumed that P is the largest prime number, so X must be a composite number; that is, it must have at least one factor other than 1 and itself.

28

A Largest Prime Number?

Let F represent the smallest prime factor of X. Because F is a prime number and P is the largest prime number, F must be a prime number less than P; that is, it must be oneof the numbers 2, 3, 4, . . . up to P.

However, it cannot be one of these numbers.

We constructed our number X in such a way that it is not divisible by 2, 3, 4, or any number up to P.

That is equivalent to saying that 2 is not a factor of X, 3 is not a factor, and so forth.

29

A Largest Prime Number?

Thus the smallest possible prime factor of X is greater than P, which is supposed to be the largest prime number.

This contradiction shows that the assumption that there is a largest prime number cannot be true.

30

Investigation B – Numbers with Personalities: Perfect and Other Numbers

A. A number n is a perfect number iff the sum of its proper divisors is equal to n. A proper divisor of a number is a divisor that is less than the number.

Discussion:

Six is the first perfect number because 1, 2, and 3 are the proper divisors of 6 and their sum is 6.

31

Investigation B – Numbers with Personalities: Perfect and Other Numbers

B. Perfect numbers lead to two other kinds of numbers, which are called abundant and deficient.

A number n is an abundant number iff the sum of its proper divisors is greater than n.

A number n is a deficient number iff the sum of its proper divisors is less than n.

Discussion:

All prime numbers are deficient by definition. Because the only proper factor of a prime number is 1, the sum of the proper factors of a prime number is 1.

cont’d

32

Investigation B – Discussion

Of the nonprime numbers less than 30, the deficient numbers are 4, 8, 9, 10, 14, 15, 16, 21, 22, 25, 26, 27.

Notice that the subset of the square numbers in the above set {4, 9, 16, 25} has only deficient numbers, although some square numbers are abundant.

Also, except for 6, any doubles of members of the set of prime numbers less than 30 {2, 3, 5, 7, 11, 13, 17, 19, 23, 29} will also be deficient.

The following numbers are abundant: 12, 18, 20, 24, and 30.

cont’d

33

Investigation B – Discussion

One pattern that students often observe that leads to a hypothesis is that all of these numbers are even. We overlooked 28 in our lists of deficient and abundant numbers.

28 is the second perfect number!

cont’d