You saw that you could devise tests for such numbers as 6 and 18 based on their relatively prime factors. Let's explore factors further.
A prime number is a number with exactly two factors. For example, the number 1 is not a prime number because it only has one factor, 1. The number 3 is a prime number because it has exactly two factors, 1 and 3. Note 4
Some numbers factor into two factors only, while others may have two factors, one or both of which can be factored further. Note 5
An important distinction can be made between the terms "factor" and "prime factor." By factors, we mean all the factors of a number. To find all the factors of 12, you can list them as shown below:
You know you can stop here because the next factor on the left would be 4, and you already have it listed on the right.
To find the prime factors of 12, you could use a factor tree:
The numbers on the bottom branch of this tree are the prime factors of 12 -- they can't be factored any further. So we say that 12 has only two prime factors, 2 and 3, and the prime factorization of 12 is 22 3. Note that we could have started the factor tree with the factors 3 and 4, and we would have derived the same prime factorization, 22 3.