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Math 952
3.2 "Prime Numbers"
Objectives:
*
Know the de…nition of a prime number.
*
Know the de…nition of a composite number.
*
Be able to list all the prime numbers less than 50.
*
Be able to determine whether a number is prime or composite.
Prime Numbers and Composite Numbers
Every counting number, (1; 2; 3; 4; 5; :::), except the number 1, has at least two factors. The following list illustrates this
idea.
Counting Numbers
Factors
Counting Numbers
7
12
Note:
Factors
13
20
In this list every number has
factors; but 7 and 13 have
factors.
De…nition:
"Prime Number"
A prime number is a whole number greater than 1 that has exactly two di¤erent factors (or divisors)
which are
:
De…nition:
"Composite Number"
kA composite number is a counting number with more than two di¤erent factors (or divisors).k
Note:
1 is neither a prime nor a composite number.
Example 1: (Prime Numbers)
Explain why 19, 31, and 41 are prime.
Example 2: (Composite Numbers)
Explain why 14, 39, and 12 are composite.
The Sieve of Eratosthenes
There is no known formula for …nding all the prime numbers. However, a famous Greek mathematician, Eratosthenes,
developed a technique that helps in …nding prime numbers. This technique is based on the concept of multiples. To …nd the
multiples of any counting number (1, 2, 3, 4, 5, 6, 7, 8,. . . ), multiply each of the counting numbers by that number. Here is
an example of multiples of 2, 3, 5, and 7:
Multiples of 2 :
Multiples of 5 :
Multiples of 3 :
Multiples of 7 :
Page: 1
Notes by Bibiana Lopez
Prealgebra by Franklin Wright
3.2
None of the multiples of a numbers, except possibly the number itself, can be prime, since each has that number as
a factor. To sift out the prime numbers according to the Sieve of Eratosthenes, we proceed by eliminating multiples, as
described in the following steps.
Step 1 :
To …nd the prime numbers from 1 to 50, list all the counting numbers from 1 to 50 in rows of ten.
Step 2 :
Start by crossing out 1 (since 1 is not a prime number). Next, circle 2 and cross out all the other multiples of 2.
Step 3 :
The …rst number after 2 that is not crossed out is 3. Circle 3 and cross out all multiples of 3.
Step 4 :
The next number that is not crossed out is 5. Circle 5 and cross out all multiples of 5.
Step 5 :
If we proceed this way, we will have the prime numbers circled and the composite numbers crossed out.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Therefore, the prime numbers less than 50 are
Facts About Prime Numbers:
1: The only even prime number is
2: All other prime numbers are
:
;
but not all odd numbers are prime.
To determine Whether a Number is Prime:
1: If the remainder is 0, then the number is composite.
2: If a quotient is smaller than the prime divisor, then the number is prime.
Example 3: (Finding prime and composite numbers)
Determine whether each of the following numbers is prime or composite by dividing by prime numbers. If the number is
composite, …nd at least two pairs of factors whose product is that number.
a) 55
b) 97
Page: 2
Notes by Bibiana Lopez
Prealgebra by Franklin Wright
c) 205
3.2
d) 719
Example 4: (Finding factors of a number)
Given the following two numbers, …nd two factors of the …rst number such that their product is the …rst number and their
sum is the second number.
a) 24; 11
b) 36; 13
c) 60;
d)
16
Page: 3
30;
7
Notes by Bibiana Lopez