is ‘n’ a prime number ?
DESCRIPTION
IS ‘N’ A PRIME NUMBER ?. DONE BY: ISHTIAQUE KHAN 1757, CSEDU. THEORY ONE. WHAT IS A PRIME NUMBER ? A Prime Number is any integral number that has exactly 2 divisors. THEORY TWO. IS 0 A PRIME NUMBER? No, 0 is not a Prime Number. IS 1 A PRIME NUMBER? No, 1 is not a Prime Number. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/1.jpg)
IS ‘N’ A PRIME NUMBER ?
DONE BY:ISHTIAQUE KHAN
1757, CSEDU
![Page 2: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/2.jpg)
THEORY ONE
• WHAT IS A PRIME NUMBER ?
A Prime Number is any integral number that has exactly 2 divisors.
![Page 3: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/3.jpg)
THEORY TWO
• IS 0 A PRIME NUMBER?
No, 0 is not a Prime Number.
• IS 1 A PRIME NUMBER?
No, 1 is not a Prime Number.
• IS 2 A PRIME NUMBER?
Yes, 2 is a Prime Number.
![Page 4: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/4.jpg)
THEORY THREE
• IF A POSITIVE INTEGER N IS NOT DIVISIBLE BY ANY NUMBER, X, IN THE RANGE:
2 <= X <= ,
THEN N IS NOT DIVISIBLE BY ANY NUMBER, X, IN THE RANGE:
< X < N
N
N
![Page 5: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/5.jpg)
LET N BE 37
37
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LET N BE 37
SO,2<=X<=SQUARE ROOT
(N)IS
2 <= X <= 6.08SINCE X IS INTEGER
2 <= X <= 6
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LET N BE 37
37 2
18.5
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LET X BE 2
![Page 8: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/8.jpg)
THEREFORE N IS NOT DIVISIBLE BY 2 OR ANY OTHER MULTIPLE OF 2
![Page 9: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/9.jpg)
LET N BE 37
2 1 2
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![Page 10: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/10.jpg)
LET N BE 37
2 2 4
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LET N BE 37
2 3 6
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![Page 12: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/12.jpg)
LET N BE 37
2 4 8
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![Page 13: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/13.jpg)
LET N BE 37
2 5 10
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![Page 14: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/14.jpg)
LET N BE 37
2 6 12
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![Page 15: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/15.jpg)
LET N BE 37
2 7 14
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![Page 16: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/16.jpg)
LET N BE 37
2 8 16
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![Page 17: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/17.jpg)
LET N BE 37
2 9 18
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![Page 18: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/18.jpg)
LET N BE 37
2 10
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![Page 19: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/19.jpg)
LET N BE 37
2 11
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![Page 20: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/20.jpg)
LET N BE 37
2 12
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![Page 21: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/21.jpg)
LET N BE 37
2 13
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![Page 22: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/22.jpg)
LET N BE 37
2 14
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![Page 23: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/23.jpg)
LET N BE 37
2 15
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![Page 24: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/24.jpg)
LET N BE 37
2 16
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![Page 25: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/25.jpg)
LET N BE 37
2 17
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![Page 26: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/26.jpg)
LET N BE 37
2 18
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![Page 27: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/27.jpg)
LET N BE 37
37 3
12.3
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LET X BE 3
![Page 28: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/28.jpg)
THEREFORE N IS NOT DIVISIBLE BY 3 OR ANY OTHER MULTIPLE OF 3
![Page 29: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/29.jpg)
LET N BE 37
3 1 3
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![Page 30: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/30.jpg)
LET N BE 37
3 2 6
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![Page 31: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/31.jpg)
LET N BE 37
3 3 9
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![Page 32: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/32.jpg)
LET N BE 37
3 4 12
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![Page 33: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/33.jpg)
LET N BE 37
3 5 15
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![Page 34: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/34.jpg)
LET N BE 37
3 6 18
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![Page 35: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/35.jpg)
LET N BE 37
3 7 21
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![Page 36: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/36.jpg)
LET N BE 37
3 8 24
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![Page 37: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/37.jpg)
LET N BE 37
3 9 27
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![Page 38: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/38.jpg)
LET N BE 37
3 10
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![Page 39: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/39.jpg)
LET N BE 37
3 11
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![Page 40: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/40.jpg)
LET N BE 37
3 12
36
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![Page 41: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/41.jpg)
LET N BE 37
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LET X BE 4
BUT 4 IS ALREADY CHECKED
![Page 42: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/42.jpg)
LET N BE 37
37 5
7.4
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LET X BE 5
![Page 43: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/43.jpg)
THEREFORE N IS NOT DIVISIBLE BY 5 OR ANY OTHER MULTIPLE OF 5
![Page 44: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/44.jpg)
LET N BE 37
5 1 5
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![Page 45: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/45.jpg)
LET N BE 37
5 2 10
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6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 46: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/46.jpg)
LET N BE 37
5 3 15
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 47: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/47.jpg)
LET N BE 37
5 4 20
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 48: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/48.jpg)
LET N BE 37
5 5 25
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 49: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/49.jpg)
LET N BE 37
5 6 30
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 50: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/50.jpg)
LET N BE 37
5 7 35
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 51: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/51.jpg)
LET N BE 37
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
LET X BE 6
BUT 6 IS ALREADY CHECKED
![Page 52: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/52.jpg)
LET N BE 37
NEXT WE HAVE ONLY PRIME NUMBERS
REMAINING. SO NON OF THEM IS A MULTIPLE OF ANY
OF THE OTHER.
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 53: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/53.jpg)
LET N BE 37
THE LEAST OF THE
REMAINING NUMBERS
IS 7
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 54: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/54.jpg)
LET N BE 37
SO THE LEAST OF THE PRODUCT OF
ANY TWO REMAINING NUMBERS IS
7 x 7 = 49WHICH IS MORE
THAN 37
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 55: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/55.jpg)
LET N BE 37
THAT MEANS ALL OTHER COMBINATION
LIKE 7 X 11 = 77
OR 29 X 29 = 841
OR 13 X 31 = 403
HAVE TO BE MORE THAN 37.
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 56: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/56.jpg)
LET N BE 37
SO, NON OF THE OTHER REMAINING NUMBERS
CAN DIVIDE 37.
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 57: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/57.jpg)
Therefore it is proof enough that 37 is not divisible by any number, X, in the range: < X < 37
If it is possible to show that 37 is not divisible by any number, X, in the range:2 <= X <= ,
That is 2 <= X <= 6
37
37
![Page 58: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/58.jpg)
So it is possible to prove ‘if an integer N is a Prime number’ by dividing it with the integers, X, in the range:
2 <= X <=
N
![Page 59: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/59.jpg)
THEORY FOUR• IF N AND X ARE TWO POSITIVE INTEGERS, THEN IN THE RANGE R1:
2 <= X <= ,
IF WE KEEP CHECKING EACH X IN THE SEQUENCE
2, 3, 4, 5, ………. ,
ALONG WITH ALL THE MULTIPLES OF X IN THE RANGE R1, THEN WHEN
X > ,
THE ONLY REMAINING UNCHECKED ELEMENTS IN R1 WILL ALL BE PRIME NUMBERS
N
N
N
![Page 60: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/60.jpg)
LET N BE 37
37
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 61: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/61.jpg)
LET N BE 37
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
THEREFORE THE RANGE R1 IS:2 <= X <= SQUARE ROOT (37)
WHICH IS
2 <= X <= 6.08
SINCE X IS INTEGRAL THEREFORE R1 IS
2 <= X <= 6
![Page 62: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/62.jpg)
LET N BE 37
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
THEREFORE THE RANGE R1 IS:2 <= X <= SQUARE ROOT (37)
WHICH IS
2 <= X <= 6.08
SINCE X IS INTEGRAL THEREFORE R1 IS
2 <= X <= 6
![Page 63: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/63.jpg)
LET N BE 37
37 2
18.5
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
LET X BE 2
![Page 64: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/64.jpg)
THEREFORE N IS NOT DIVISIBLE BY 2 OR ANY OTHER MULTIPLE OF 2
![Page 65: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/65.jpg)
LET N BE 37
2 1 2
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 66: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/66.jpg)
LET N BE 37
2 2 4
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 67: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/67.jpg)
LET N BE 37
2 3 6
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 68: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/68.jpg)
LET N BE 37
NOW, X = 3THEREFORE,
SQUARE ROOT (6) < XTHAT IS2.44 < 3
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 69: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/69.jpg)
LET N BE 37
NOW THE ONLY NUMBERS REMAINING
IN THE RANGE R1:
2 <= X <= 6
ARE PRIME NUMBERS:3 AND 5
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
![Page 70: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/70.jpg)
LET N BE 401
401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
![Page 71: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/71.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
THEREFORE THE RANGE R1 IS:2 <= X <= SQUARE ROOT (401)
WHICH IS
2 <= X <= 20.02
SINCE X IS INTEGRAL THEREFORE R1 IS
2 <= X <= 20
![Page 72: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/72.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
THEREFORE THE RANGE R1 IS:2 <= X <= SQUARE ROOT (401)
WHICH IS
2 <= X <= 20.02
SINCE X IS INTEGRAL THEREFORE R1 IS
2 <= X <= 20
![Page 73: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/73.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
401
2200.5
LET X BE 2
![Page 74: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/74.jpg)
THEREFORE N IS NOT DIVISIBLE BY 2 OR ANY OTHER MULTIPLE OF 2
![Page 75: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/75.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 1 2
![Page 76: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/76.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 2 4
![Page 77: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/77.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 3 6
![Page 78: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/78.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 4 8
![Page 79: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/79.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 5 10
![Page 80: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/80.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 6 12
![Page 81: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/81.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 7 14
![Page 82: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/82.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 8 16
![Page 83: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/83.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 9 18
![Page 84: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/84.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 10
20
![Page 85: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/85.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
LET X BE 3
401
3133.7
![Page 86: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/86.jpg)
THEREFORE N IS NOT DIVISIBLE BY 3 OR ANY OTHER MULTIPLE OF 3
![Page 87: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/87.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
3 1 3
![Page 88: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/88.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
3 2 6
![Page 89: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/89.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
3 3 9
![Page 90: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/90.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
3 4 12
![Page 91: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/91.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
3 5 15
![Page 92: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/92.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
3 6 18
![Page 93: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/93.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
NOW, X = 5THEREFORE,
20 < XTHAT IS4.47 < 5
![Page 94: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/94.jpg)
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
NOW THE ONLY NUMBERS REMAINING
IN THE RANGE R1:
2 <= X <= 20
ARE PRIME NUMBERS:5, 7, 11, 13, 17 AND 19
![Page 95: IS ‘N’ A PRIME NUMBER ?](https://reader035.vdocuments.us/reader035/viewer/2022062800/5681417c550346895dad6bc2/html5/thumbnails/95.jpg)
Thank you!