100 with an easy method. This method was given by the Greek Mathematician Eratosthenes, in
Without actually checking the factors of a number, we can find prime numbers from Ito
the third century BC. Let us see the method. List all the numbers from 1 to 100, as shown below:
8
9
10
5
18
19
20
4
17
2
3
16
15
14
28
29
30
11
27
13
12
26
25
39
24
38
40
21
22
23
37
36
35
34
48
49
33
31
50
32
47
46
45
44
41
43
59
42
58
60
57
56
55
51
52
54
53
68
69
70
66
67
65
61
62
63
64
78
79
80
76
77
75
71
72
74
73
89
88
87
90
86
81
83
85
82
84
97
98
99
100
91
96
92
93
94
95
Step-1: Cross out 1 because it is neither prime nor composite.
Step-2: Encircle 2, cross out all the other multiples of 2, i.e. 4, 6, 8 and so on.
Step-3: You will find that the next uncrossed number is 3. Encircle 3 and cross out all the o
multiples of 3.
kep-4: The next uncrossed number is 5. cross out all the othEncircle 5 ander multiples of 5
ep-5: Continue this process till all the numbers in the list are either encircled or crossed ou
All the encircled numbers are prime numbers. All the crossed out numbers, other
e composite numbers
Answers
Step-by-step explanation:
A prime is an integer greater than 1 that is only divisible by 1 and itself. The integers 2, 3, 5, 7, 11 are prime integers. Note that any integer greater than 1 that is not prime is said to be a composite number.
The Sieve of Eratosthenes
The Sieve of Eratosthenes is an ancient method of finding prime numbers up to a specified integer. This method was invented by the ancient Greek mathematician Eratosthenes. There are several other methods used to determine whether a number is prime or composite. We first present a lemma that will be needed in the proof of several theorems.
Every integer greater than one has a prime divisor.
We present the proof of this Lemma by contradiction. Suppose that there is an integer greater than one that has no prime divisors. Since the set of integers with elements greater than one with no prime divisors is nonempty, then by the well ordering principle there is a least positive integer n greater than one that has no prime divisors. Thus n is composite since n divides n . Hence
n=abwith 1<a<nand 1<b<n.(2.1.1)
Notice that a<n and as a result since n is minimal, a must have a prime divisor which will also be a divisor of n .
If n is a composite integer, then n has a prime factor not exceeding n−−√ .
Since n is composite, then n=ab , where a and b are integers with 1<a≤b<n . Suppose now that a>n−−√ , then
n−−√<a≤b(2.1.2)
and as a result
ab>n−−√n−−√=n.(2.1.3)
Therefore a≤n−−√ . Also, by Lemma 3, a must have a prime divisor a1 which is also a prime divisor of n and thus this divisor is less than a1≤a≤n−−√ .
We now present the algorithm of the Sieve of Eratosthenes that is used to determine prime numbers up to a given integer.