Prime Factors

Definition of A Factor 

The word ‘Prime Factors’ is made up of two distinct terms, Prime and Factor, both of which are important in Mathematics. To understand the concept of prime factors properly, we need to understand what are factors in detail and then, move on to understand what are prime numbers. When we multiply two numbers, we get the product. Factors are numbers which have been multiplied to get the product. Likewise, factors can divide the product perfectly with no reminder. 

Let us take a simple example:


If we take a look at number 16, then to get the number 16, we would have to multiply 8 with 2. So, in this case, 8 and 2 are factors of 16. On the other hand, if we divide 16 by 2, i.e., 16/2, we get the remainder of 0. So, a number when divided by any of its factors would always give us the remainder of 0 - that is, it will be perfectly divisible. One product can have multiple factors as well like 16 can also have factors 4,4 (4 x 4) and 16,1 (16 x 1).


What is a Prime Number

Now that you know what factors are, let us see the meaning of a prime number. A prime number is a number, which only has two factors: 1 and the number itself. 2 is a prime number, and 4 is not a prime number. This is because, 2 = 2 x 1 but, 4 = 2 x 2. So, 2 can only have two factors, itself and 1. 

Also, the numbers which are a combination of prime numbers are called composite numbers. So, 4 is a composite number whereas 2 is a prime number (in fact, 2 is the only prime number which is even. All other even numbers are divisible by 2.)


What are the Prime Factors or Prime Factors Definition

Prime factors are simply factors of a number which are prime numbers. Here is an example:

Take the number 8. 8 can have different factors, that is,  8 = 4 x 2 = 2 x 2 x 2 = 8 x 1

So, we can say that the numbers 4, 2, 8, 1 are the factors of 8. But the prime factor of 8 is only 2 (because 1 is neither prime nor a composite number). 4 in itself is not a prime number (4 = 2 x 2) and neither is 8 (8 = 4 x 2).

So, prime factor meaning tells us that a number which is both prime and a factor of any given number is a prime factor. 


How to Find Prime Factors of a Number

The method of finding prime factors of any given number is called prime factorization. What is the meaning of prime factorization?

Prime factorization is the process in which we write any number in the form of its prime factors. Now there are two different ways to find the prime factors of a number; the Factor tree method and the repeated division method. 


How To Find Prime Factorization of A Number Using the Factor Tree Method

The method of factor tree is straightforward. Take the number, and then, with two arrows representing the branches of a tree, you break the number into any of the factors. You do this process until you reach the prime factors. 

Example 1: 


Find the prime factors of 36 using factor tree method.

Solution:


We see that the prime factors of 36 are 2 and 3. So, we can write the prime factorization of 36 as: 

36 = 2 x 2  x 3 x 3.

One benefit of using the factor tree is that this method is graphical in how it represents the breakdown into factors. However, reaching the prime factors  can be time-consuming. 


How to Calculate Prime Factors using the Repeated Division Method?

As the name suggests, in this method, we begin by dividing the number with its smallest prime factor, and we keep on dividing the resultant quotient with its smallest prime factor until we reach 1 as the final quotient. Here’s an example.

Finding prime factors of 36 using repeated division method:

In this case, we divide 36 by the prime factor of 2 and get the resultant quotient 18, which we divide with 2 again. We keep on dividing until we get the number 1.

So, 36 = 2 x 2  x 3 x 3

The prime factors of 36 are 2 and 3.

FAQ (Frequently Asked Questions)

1. How should I begin the method of repeated division to find out prime factors of any given number?

You should begin this method by choosing any prime number which is the factor of that number. So, in the example of finding the prime factors of 36, we can see that the number is even. So, if the number is even that means, one of the prime factors would be two. So, we begin the process by dividing the number by 2 which is also the lowest prime number. After division, we find the number 18. 18 is again an even number, so we can still divide it by 2 to continue our process. Then we encounter 9. 3 is the lowest prime factor of 9. So, we divide 9 by 3. Then we are left with 3 itself, meaning another division by 3 would make the quotient 1. You can use divisibility tests to find the lowest prime factor.

2. What is the best method to use the factor tree or repeated division?

You can use any method in any question and get the prime factors. However, we prefer to use the method of repeated division for larger numbers. The main reason why we prefer that method is because it is relatively easier to keep track of the factors and it is less time-consuming. You can find all the prime factors of the number on the left-hand margin easily. One major disadvantage of the factor tree method is that the amount of space each answer takes. You can definitely draw out a tree for a small number. But when the number is bigger, the branches increase, and the chances to make an error also increases.