Are there any steps to help determine if a number is prime or composite?

Answer 1

It is sometimes not easy, especially if the (natural) number is long. But there are a few easy checks you can make that help. The fastest way is to find an online site that answers the question for numbers you submit, quickly, easily and presumably accurately. If you cannot access a computer, the checks below amount to being filters for some kinds of composite numbers. If you can show that the number in question (niq) meets one of these criteria, you have demonstrated that it is composite. Have a calculator handy.

If the niq (of 2 or more digits, understood throughout) ends with any of these digits: 0, 2, 4, 5, 6, or 8, then you KNOW that it is composite. If it ends with 0, 2, 4, 6, or 8, you know that it is even and can be divided by 2. If it ends in 0 or 5, you know it can be divided by 5.

Calculate the digital root* of the niq. This is fun and easy to learn. Just add all the digits together. When you get an answer of 2 digits, add them together and keep going. The single digit you end up with is the digital root of the niq.

*If the digital root is equal to 9, then you KNOWthat the niq is composite, and it has at least a factor of 9 besides itself and one. This is the only digital root (9) that can indicate absolutely that the niq is composite; the other roots can happen for primes and nonprimes alike.

Divide the number by 6 and look at the digital remainder^, the digits to the right of the decimal point. Remainders after dividing a natural number by 6 must be one of the following values: .0 exactly, .3333 repeating, .5 exactly, .6666 repeating, .8333 repeating, or .16666 repeating.

^If the digital remainder has only one digit in it, a single time or repeating, (.0, .3333, .5, .6666) you KNOW the niq is composite. Any other remainder and you cannot be certain either way.

If you are still in doubt about the niq, then you have to start dividing to see if you can find factors. You don't have to divide by even numbers because if there is no remainder the niq is even and you have already eliminated it. You don't have to divide by any number larger than the square root of the niq. If any number larger than the square root is a factor of the niq, then by the time you get to the square root, you will already have found the smaller factor that goes with it.

Divide by 3.

Divide by 7. (You have already eliminated even numbers and numbers divisible by 5).

Divide by 11. (8 and 10 are even, and you would already know from test #2 if the niq is a multiple of 9).

Divide by 13, then by 17 (you've already eliminated multiples of 5 so no need to divide by 15 or any other multiple of 5). Keep working your way up.

If there are no divisions without remainders, the niq is prime.