- Prime Factoring
- Factoring is an idea you might be familiar with from multiplication.
Numbers that can be multiplied together to get another number are
its factors. For example, 4*3 = 12, so 3 and 4 are factors of 12.
However, they're not its only factors; 1, 2, 6, and 12 are other
factors of 12. (Another way of defining a factor is a number that
goes evenly into the number you're factoring.)
- A number is prime if it can not be divided evenly by anything
except itself and 1. For example, 5 is a prime number, because
the only factors of 5 are 1 * 5 = 5. However, 12 is not a prime
number, because 1 * 12 = 12, 2 * 6 = 12, and 3 * 4 = 12. Prime
factorization means finding all the prime numbers that are
factors of a number.
- It's helpful to have in your head the divisibility rules for prime
numbers like 2, 3, 5, and 7. You'll find them in the Dr. Math FAQ:
- Divisibility Rules
2 If the last digit is even, the number is divisible by 2.
3 If the sum of the digits is divisible by 3, the number
4 If the last two digits form a number divisible by 4,
the number is also.
5 If the last digit is a 5 or a 0, the number is divisible
6 If the number is divisible by both 3 and 2, it is also
divisible by 6.
7 Take the last digit, double it, and subtract it from the
rest of the number; if the answer is divisible by 7
(including 0), then the number is also.
8 If the last three digits form a number divisible by 8,
then the whole number is also divisible by 8.
9 If the sum of the digits is divisible by 9, the number
10 If the number ends in 0, it is divisible by 10.
- A good basic knowledge of the multiplication table is also a help,
together with familiarity with smaller prime numbers (2,3,5,7,11,13,17,19).
- Let's take an example. How would you find the prime factorization
of 126? Well, one way you could start would be by noticing that
126 is even. 2 is the only even prime number, and it divides evenly
into every even number. So, if we divide 126 by 2 we get 63.
- 126 = 2 x 63
- Next we know from our multiplication tables that 63 = 7 x 9.
- 126 = 2 x 7 x 9
- We know that 7 is prime; what about 9? Nine is not prime: 9 = 3 x 3.
- 126 = 2 x 7 x 3 x 3
- Now we can stop since we have reached only prime numbers. The prime
factors of 126 are: 2 x 3 x 3 x 7, or 2 x 3^2 x 7.