is ‘n’ a prime number ?

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IS ‘N’ A PRIME NUMBER ? DONE BY: ISHTIAQUE KHAN 1757, CSEDU

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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 Presentation

TRANSCRIPT

Page 1: IS ‘N’ A PRIME NUMBER ?

IS ‘N’ A PRIME NUMBER ?

DONE BY:ISHTIAQUE KHAN

1757, CSEDU

Page 2: IS ‘N’ A PRIME NUMBER ?

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 ?

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 ?

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 ?

LET N BE 37

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Page 6: IS ‘N’ A PRIME NUMBER ?

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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Page 7: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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LET X BE 2

Page 8: IS ‘N’ A PRIME NUMBER ?

THEREFORE N IS NOT DIVISIBLE BY 2 OR ANY OTHER MULTIPLE OF 2

Page 9: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 10: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 11: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 12: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 13: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 14: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 15: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 16: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 17: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 18: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 19: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 20: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 21: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 22: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 23: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 24: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 25: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 26: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 27: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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LET X BE 3

Page 28: IS ‘N’ A PRIME NUMBER ?

THEREFORE N IS NOT DIVISIBLE BY 3 OR ANY OTHER MULTIPLE OF 3

Page 29: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 30: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 31: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 32: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 33: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 34: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 35: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 36: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 37: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 38: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 39: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 40: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 41: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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LET X BE 4

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Page 42: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 43: IS ‘N’ A PRIME NUMBER ?

THEREFORE N IS NOT DIVISIBLE BY 5 OR ANY OTHER MULTIPLE OF 5

Page 44: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 45: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 46: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 47: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 48: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 49: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 50: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 51: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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Page 52: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

NEXT WE HAVE ONLY PRIME NUMBERS

REMAINING. SO NON OF THEM IS A MULTIPLE OF ANY

OF THE OTHER.

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Page 53: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

THE LEAST OF THE

REMAINING NUMBERS

IS 7

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Page 54: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

SO THE LEAST OF THE PRODUCT OF

ANY TWO REMAINING NUMBERS IS

7 x 7 = 49WHICH IS MORE

THAN 37

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Page 55: IS ‘N’ A PRIME NUMBER ?

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.

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Page 56: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

SO, NON OF THE OTHER REMAINING NUMBERS

CAN DIVIDE 37.

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Page 57: IS ‘N’ A PRIME NUMBER ?

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

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Page 58: IS ‘N’ A PRIME NUMBER ?

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 ?

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 ?

LET N BE 37

37

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Page 61: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

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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 ?

LET N BE 37

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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 ?

LET N BE 37

37 2

18.5

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LET X BE 2

Page 64: IS ‘N’ A PRIME NUMBER ?

THEREFORE N IS NOT DIVISIBLE BY 2 OR ANY OTHER MULTIPLE OF 2

Page 65: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

2 1 2

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Page 66: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

2 2 4

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Page 67: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

2 3 6

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Page 68: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

NOW, X = 3THEREFORE,

SQUARE ROOT (6) < XTHAT IS2.44 < 3

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Page 69: IS ‘N’ A PRIME NUMBER ?

LET N BE 37

NOW THE ONLY NUMBERS REMAINING

IN THE RANGE R1:

2 <= X <= 6

ARE PRIME NUMBERS:3 AND 5

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Page 70: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

401

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Page 71: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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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 ?

LET N BE 401

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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 ?

LET N BE 401

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401

2200.5

LET X BE 2

Page 74: IS ‘N’ A PRIME NUMBER ?

THEREFORE N IS NOT DIVISIBLE BY 2 OR ANY OTHER MULTIPLE OF 2

Page 75: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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2 1 2

Page 76: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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2 2 4

Page 77: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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2 3 6

Page 78: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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2 4 8

Page 79: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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2 5 10

Page 80: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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2 6 12

Page 81: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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2 7 14

Page 82: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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2 8 16

Page 83: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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2 9 18

Page 84: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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Page 85: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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LET X BE 3

401

3133.7

Page 86: IS ‘N’ A PRIME NUMBER ?

THEREFORE N IS NOT DIVISIBLE BY 3 OR ANY OTHER MULTIPLE OF 3

Page 87: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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3 1 3

Page 88: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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3 2 6

Page 89: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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3 3 9

Page 90: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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3 4 12

Page 91: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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20

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3 5 15

Page 92: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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3 6 18

Page 93: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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20

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397

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NOW, X = 5THEREFORE,

20 < XTHAT IS4.47 < 5

Page 94: IS ‘N’ A PRIME NUMBER ?

LET N BE 401

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397

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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 ?

Thank you!