Eratosthenes(276 - 194 B.C.) was a Greek mathematician who is famous for measuring the diameter of Earth, but also for his work on prime numbers. Prime numbers are very unique. What are they?
A prime number is a whole number, greater than one, which can only be divided by one and itself. In other words, it has exactly two factors.
A composite number is a whole number greater than one, which has more than two factors.
Think about it.
Are zero and one prime or composite numbers?
Neither.
Neither. Zero is unique in that it cannot be divided by itself. Remember that division by zero is undefined.
For example, let’s try 0 ÷ 5 and 5 ÷ 0.
The definition of division says that the number you are dividing by and the number you get for the solution must multiply to give the number you divided into. In other words,
This is true because 5(0) = 0.
So, we may divide into zero, but not by zero. Division by zero is not defined and zero is neither prime nor composite.
1 has exactly one factor, itself. 1(1) = 1. So, it is not prime or composite.
Zero and One
Zero is neither prime nor composite.
0 ÷ a = 0
a ÷ 0 is undefined
One is neither prime nor composite.
Example 1: Listing Factors and Identifying
List the factors of each number and identify if the number is prime or composite.
36
One way to find all factors is to list them in sets that multiply to give you the number.
1 x 36
2 x 18
3 x 12
4 x 9
5 x no number
6 x 6
At this point, you are out of numbers that will work.
The factors are {1, 2, 3, 4, 6, 9, 12, 18, 36}.
So, 36 is composite.
You need to list six only one time.
15
1 x 15
2 x no number
3 x 5
4 x no number
5 x 3
Since you have repeated one of the numbers, 3, you are finished finding the factors.
The factors are {1, 3, 5, 15}.
So, 15 is composite.
19
1 x 19
2 x no number
3 x no number
4 x no number
5 x no number
etc.
There are only two factors, {1, 19}.
So, 19 is prime.
Activity 1 (10 points)
Answer each of the problems below.
Eratosthenes invented a method, the Sieve of Eratosthenes, for efficiently constructing tables of prime numbers. This is his method.
Make this list on a piece of scrap paper. There are 10 numbers in each row and 10 rows. It is all the whole numbers from 1 to 100.
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
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
Cross one (1) off the list because you know it is not prime (since it only has one factor, itself).
Next, draw a box around two (2). Then cross off all multiples of two. (A multiple of two is any number that two will divide into evenly; therefore all the multiples of two are composite numbers. You know that two can divide into any even number.)
Box the next unmarked number in the list, which is three (3). Then cross off all multiples of three since they are composite numbers.
Box the next unmarked number, five (5). Then cross off any multiple of five in the list.
Continue this process for the next unmarked number, 7. Keep going until you reach 19. Then box all remaining unmarked numbers to 100.
When you have finished, click "Polynomials" and find the 7-Activity 1 link to answer the questions.This activity is meant to help you prepare for 7-Activity 1. You may take 7-Activity 1 only one time, so check your understanding of the material before taking it.
A prime number is a whole number, greater than one, which can only be divided by one and itself. In other words, it has exactly two factors.
Think about it.
Activity 1 (10 points)
