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Prepared by:  RONIETA T. VILLANUEVA 			   MAT - MATHEMATICS

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Even Numbers – Any number that can be divided by 2 Odd Numbers – Any number that cannot be divided by 2 Composite Number – An integer that can be divided by at least one other number (a factor) other than itself Factor – a number that can divide another number without a remainder.

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Prime numbers  are the positive integers having only two factors, 1 and the integer itself. If p is a prime, then it’s only factors are necessarily 1 and p itself. Any number which does not follow this is termed as composite numbers, which means that they can be factored into other positive integers.

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For example, factors of 6 are 1,2,3 and 6, which are four factors in total. But factors of 7 are only 1 and 7, totally two. Hence, 7 is a prime number but 6 is not, instead it is a composite number. But always remember that 1 is neither prime nor composite.

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The prime number was discovered by Eratosthenes (275-194 B.C., Greece). He took the example of a sieve to filter out the prime numbers from a list of natural numbers and draining out the composite numbers.

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2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

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The following two methods will help you to find whether the given number is a prime or not.

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We know that 2 is the only even prime number. And only two consecutive natural numbers which are prime are 2 and 3. Apart from those, every prime number can be written in the form of 6n + 1 or 6n – 1 (except the multiples of prime numbers, i.e. 2, 3, 5, 7, 11), where n is a natural number.

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6(1) – 1 = 5 6(1) + 1 = 7 6(2) – 1 = 11 6(2) + 1 = 13 6(3) – 1 = 17 6(3) + 1 = 19 6(4) – 1 = 23 6(4) + 1 = 25 (multiple of 5) …

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To know the prime numbers greater than 40, the below formula can be used. n ²  + n + 41, where n = 0, 1, 2, ….., 39…

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(0)2 + 0 + 0 = 41 (1)2 + 1 + 41 = 43 (2)2 + 2 + 41 = 47 …..

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Solution: No, because it can be divided evenly by 2 or 5, 2×5=10, as well as by 1 and 10. Alternatively, Using the method 1, let us write in the form 6n ± 1.  10 = 6(1) + 4 = 6(2) – 2 This is not of the form 6n + 1 or 6n – 1. Hence, 10 is not a prime number.

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Solution: Let us write the given number in the form of 6n ± 1. 6(3) + 1 = 18 + 1 = 19 Therefore, 19 is a prime number.

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Example 3: Check if 64 is a prime number or not.

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Solution: The factors of 64 are 1, 2, 4, 8, 16, 32, 64. Hence, it is a composite number and not a prime number.

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Zero and 1 are not considered prime numbers. The only even prime number is 2. All other even numbers can be divided by 2. The number               is the largest prime number with 24,862,048 digits (found in 2018).

PRIME NUMBERS

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