How to Determine the Prime Factorization of a Number
There are two main ways for determining the prime factors of a number. I will demonstrate both methods, and let you decide which you like best.
Both methods start out with a factor tree. A factor tree is a diagram that is used to break down a number into its factors until all the numbers left are prime.
The first way you can use a factor tree to find the factorization of a number is to divide out prime numbers only. Let's factor 24 using this method.
Since 24 is an even number, the first prime number that can be factored out is a 2. This leaves us with 2 × 12. Again, 12 is an even number, so we can factor out another 2, leaving us with 2 × 2 × 6. Since 6 is even, we can factor out a third two, leaving 2 × 2 × 2 × 3. All of these numbers are prime, so the factorization is complete.
The other method for using a factor tree to find the prime factorization of a number is just to pull out the first factors that you see, whether they are prime or not. Looking back at our example from above, let's factor 24 again using this method.
The first thing you might notice is that 6 * 4 is 24, so that is one set of factors for 24. Since neither of these numbers are prime, we can continue to factor both of them. 6 can be broken down to 2 * 3, and 4 can be broken down to 2 * 2. Now all of our factors are prime, and the factorization of 24 is complete, again giving the answer of 2 * 2 * 2 * 3.
Both of these methods work equally well, and can be used interchangeably. There are people who like to use certain tricks to pull out prime numbers first without having to decide what other numbers might be factors of the original number.
The tricks to find some of the prime numbers are:
- Any even number is divisible by 2.
- If you add up the digits in a large number and the sum you get is divisible by 3, the number is also divisible by 3.
- A number that ends with a 5 or 0 is divisible by 5.
These little tricks can help you factor larger numbers where it might not be easily apparent where to start.
Let's try another example.
Find the prime factors of 117.
The first thing I notice about this number is that if you add the digits (1+1+7), you get 9. This means that the number is divisible by 3. Since it is not even, and does not end with a 5 or 0, it is not divisible by 2 or 5, so we can start with the 3.
117 divided by 3 is 39, so our first two factors are 3 and 39. 39 is also divisible by three because 3 + 9 = 12
39/3 equals 13
13 is a prime number, so our factorization is complete, and the factors of 117 are 3 × 3 × 13.