Arithmetic – Evaluation of Prime Numbers

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Prime Number and Prime Factor

• Prime Number: A natural number larger than 1 which cannot be

expressed as the product of two smaller numbers.

• E.g: 2, 3, 5, 7, 11, 13, 17, 19.

• Primer Factor: Atomic Elements of natural number multiplication.

• E.g: 12 = 2 * 2 * 3

• In the above multiplication 2 or 3 are prime factors and cannot be split

further.

• Every number is a prime number of product of prime numbers.

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Infinity of Prime numbers

• Currently there is no end to the list of the prime numbers.

• Euclid’s Proof:

• Lets think that there are only finite prime numbers with Pn be the last prime

number.

• Let P1 , P2, P3 …. Pn be the list of the prime numbers ever known.

• Multiply of all the prime numbers and product is

• P = p1 * p2 * .. Pn.

• Add 1 to this number P+1. P+1 is a natural number which is greater than 1

and doesn’t have any prime factors.

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Sieve of Eratosthenes

• How to find whether a number is a prime?

• Is Even number ?

• Is divided by small prime numbers [3 or 5 or 7 or 11]?

• Let the given number be N. Find the closest root of N and name it as n.

• Find out whether N is divided by any of the prime numbers below n

• 2 , 3 , 5, 7, 11, 13 , 17 … n

• If none of the above steps doesn’t strike off the number N. Then N is a prime

number.

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Examples

• Check whether 1231 is a prime number ?

Yes

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Prime Factorization

• Also called as Fundamental Theorem of arithmetic

• Any given integer is prime by itself or a product of prime numbers

[factors].

• The product is unique up to the order of the factors.

• E.g: Find the prime factor product for 900

• 900 = 9 * 100

= 3 * 3 * 5 * 5 * 2 * 2 [2 ,3 ,5 are prime numbers. Hence this is the final

product in the order of prime factors]

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Twin Primes and Prime Numbers below 100

• Twin prime is defined as pair of prime numbers whose difference is 2.

• Let p be the prime number then [P-2 , P] or [P, P+2] as Twin primes if P-2 or

P+2 is also a prime number

• E.g: [11,13] [17,19], [29,31] ..

• List of Prime Numbers below 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

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Examples

• Quantity A: Smallest prime number multiplied by 5 and divided by

least common multiple of 3 and 5

Quantity B: Smallest prime number multiplied by 3 and divided by the

greatest common factor of 9 and 12

Ans:

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Examples

• Quantity A: Number of prime numbers between 0 and 100

Quantity B: Number of prime numbers between 100 and 200

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Examples

• A: Sum of all prime factors of 144?

B: Product of all prime factors of 12?

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