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.