Prime Numbers and Prime Factorization

Strictly speaking a prime number is a natural number (whole number greater than 0) that is not divisible by any other number than itself and 1. 1 is not a prime, so the first prime number is 2 (2 can only be divided with 1 and 2).

This is the general rule and it goes for any natural number out there. If you cannot divide the number with anything but 1 and itself, it's a prime. If the number is not a prime, then it's composite. Let's look at some examples.

12 is a natural number, and if you divide it by 2 you get 6. Therefore 12 is not a prime number, but it is a composite number.

13 on the other hand is not divisible by any number except 1 and 13. 13 is therefore a prime number.

Other examples of prime numbers, beginning with the smallest are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 and so on.

In another article I covered factorizations. To find the prime factorization of a number you divide the number by the lowest prime (2) and see if there is a remainder. If the remainder is zero, you have found the prime factorization for that number. If there was a remainder you try the next prime number which is 3, and keep going until you have a zero remainder. All of this can be done without a calculator, as long as you know division.

If you find one prime factor of a number, but the other factor is a composite you will have to break the composite number down into primes just as you did the original number. Prime factorization should only consist of primes, so you might have to go through several steps.

One example is 220. You start by dividing it by 2 and end up with the factors 2 and 110. 110 is a composite number, so you divide 110 with 2 to end up with 55. 55 is also a composite number so you divide 55 with 5 and end up with 11. The prime factorization of 220 is thus 2*2*5*11.

One thing that is quite unique with primes is that there is only one prime factorization of a number. This means that 2*2*5*11 is the only prime factorization of 220. However, if you just wanted to find a factorization, and it did not matter if it was prime or composite there could be several possibilities such as 110*2, 2*10*11, 20*11 and so on.