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Given: All Prime numbers are odd, If a number is even it is not prime, If a number is odd and it can be divided by something other than itself or 1 it is not prime.

1.) One way to test an odd number and determine whether it is prime is to start with the number 3 and try dividing it perfectly into the odd number:

For example, 27 is odd, but it can be divided by the number 3 which divides perfectly into 27 nine times. 3 x 9 = 27. So 27 is not prime.

If 3 doesn't divide perfectly into the number then keep adding 2 to the factor and try dividing the number by the new factor:

For example, 91 is odd, but 3 does not perfectly divide 91, so we add 2 to 3 and try dividing 91 by 5 which we know immediately will not work since 91 does not end in 0 or 5, so we add 2 more and try dividing 91 by 7. We find that 7 divides perfectly into 91 thirteen times. 7 x 13 = 91, so 91 is not prime.

If you know your times tables it makes this determination easier since we can also eliminate all multiples of each number that doesn't divide perfectly into the test number. For example, since 2 doesn't divide perfectly into a Prime number then all multiples of 2 (all even numbers) are not eligible as potentially divisors. If 3 doesn't divide perfectly then all multiples of 3 do not have to be tested (i.e., 9, 15, 21, 27, 33...)

Last example using the above tests: Is 97 prime?

Given: When I use the word "divided" I mean perfectly divided

Is 97 even? No, so all multiples of 2 are eliminated from testing

Can 97 be divided by 3? No, so all multiples of 3 are eliminated from testing

Can 97 be divided by 5? No, so all multiples of 5 are eliminated from testing

Can 97 be divided by 7? No, so all multiples of 7 are eliminated from testing

Can 97 be divided by 11? No, so all multiples of 11 are eliminated from testing

Can 97 be divided by 13? No, so all multiples of 13 are eliminated from testing

Can 97 be divided by 17? No, so all multiples of 17 are eliminated from testing

Can 97 be divided by 19? No, so all multiples of 19 are eliminated from testing

Can 97 be divided by 23? No, so all multiples of 23 are eliminated from testing

Can 97 be divided by 29? No, so all multiples of 29 are eliminated from testing

Can 97 be divided by 31? No, so all multiples of 31 are eliminated from testing

We can safely say that 97 is prime at this point. Why? Because the next number to test is 37 and since 37 x 3 is 111 and we cannot divide 97 by 3 then no larger number will work. All factors have been effectively tested, albeit the slow way, but it works 100% of the time to determine primeness.

Larger numbers may cause this method to be too time consuming; a formula used with a calculator would best be implemented to calculate primeness with larger numbers. You could also print out a prime number table as a reference.

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