Common Factors

Before going to understand about common factors, let's recall what a factor is? When a number is multiplied with another number and gives the given number as a result, then the number which we multiplied is known as a factor.


So, coming to common factors, a number which gets divided into two numbers with zero as it's a reminder, then we call it a common factor. When do we say common? If anything has a similarity, then they are common. 


Definition:- 

Common Factors definition is - they can be defined as the factors which can have the same for two or more numbers, and then these factors are termed as common factors.

Thus, when a number is divided exactly for two or more numbers, then that number is called a common factor for those numbers.


Properties:-

  • Factors should not be greater than the given number.

  • Factors should be less than or equal to the given number.

  • Numbers may have multiple common factors.

  • Every number has a minimum of two factors.

  • 1 and itself are the basic factors for every number. So, 1 is a common factor for all sets.

  • Shouldn't write repeated numbers as two factors.


Finding Common Factors:-

Common Factors can be calculated for two numbers or more than two numbers. To find the common factors with examples we need to follow these steps,

  • Take the two numbers separately.

  • Find the factors of each number and write to them.

  • Now compare and find which numbers are common in both factors.

  • Those are the common factors for the given two numbers.

  • The same procedure can be followed by more numbers also.


Thus, these are the steps we need to follow to find the common factors for a given set of numbers.


If we get three or four common factors for a set of numbers, then we will treat the big number as the highest common factor.  Below are a few examples of finding common 

factors.


Examples of common factors:-

To understand more clearly and easily, let us start finding common factors for a few examples. First, we will find two numbers.


Common Factors of 6, 12:-

First, factors of 6 are 1,2,3,6.   

Then we need to find,

 factors of 12 are 1,2,3,4,6,12.

 Now, the common factors are 1,2,3,6.


Common Factors of 15,25:-

First, factors of 15 are 1,3,5,15.

Next, factors of 25 are 1,5,25.

Now the common factors are 1,5.

Thus, this is the process to be followed to find the common factors and their examples.

Let's see the examples for a set of three,


Common Factors of 12,24,48.

First factors of 12 are 1,2,3,4,6,12.

Factors of 24 are 1,2,3,4,6,8,12,24.

Factors of 48 are, 1, 2, 3,4,12,16,24,48.

Now let's find common factors of 12,24,48 are, 1,2,3,4.


Common Factors of 10,20,25:-

First factors of 10 are 1,2,5,10.

Factors of 20 are, 1,2,4,5,10,20.

Factors of 25 are 1,5,25.

So, the common factors are 1,5.


Common Factors of 12,15,18,24:-

Factors of 12 are, 1,2,3,4,6,12.

Factors of 15 are, 1,3,5,15.

Factors of 18 are, 1,2,3,6,9,18.

Factors of 24 are 1,2,3,4,6,8,12,24.

Thus, common factors are 1,3.


In the same manner, we can find the common factors for n numbers in a set too.

In this method, if we can move further, we are also able to find the highest common factor. This can be found by different methods. 


Among all the obtained common factors, we just need to pick a large number. This must be termed as the highest common factor. This also can be found by using the prime factorization method.


Advantages of finding common factors:-

 Finding common factors can help us,

  • To simplify complex fractions.

Like, if we have,

               25/625 = 5/25= ⅕

              96/144 = 8 /12= ⅔

  • To represent big fractions also in a multiplication way with the greatest factor.

Like, if we have 

       144 = 12 *12

        65 = 5 *13

Thus, we have these kinds of applications by using common factors.


Conclusion:-

Hence, the common factors are easy to find but have many applications. So everyone should learn and make the best use of it. Thus, the definition of the common factor involves that a number that divides exactly another set of numbers without leaving any reminder is nothing but common factors.

It is very helpful to solve complex equations in mathematics which is a part of our daily life. It is a simple basic rule used in the multiplication process which has further extensions to solve numerous sums. In the competitive exams also, 8t has a significant role for students to save their time and to crack answers fastly.


FAQ (Frequently Asked Questions)

1. What is a common factor? What is it's properties?

Ans. A Common Factor is a number that divides two or more numbers with a remainder zero; then, it is the common factor for the given numbers. We may have one or more than one as common factors. We have a certain extension like the greatest common factor. The properties are mostly similar to the properties of factors like,

  • It should be a finite number.

  • May have one or more common factors.

  • We can find a set of two or more numbers.

  • 1 is the minimum and basic common factor for any set.

  • Should have integers to get common factors.

  • Zero may not have factors.


2. What is HCF? How is it used?

Ans. HCF means the highest common factor. It is easy to find. After finding out all the common factors, we need to identify the large number, which is called the highest common factor. It is noted as HCF. 

It is used by children to solve the biggest fractions and also helpful to find the least common multiple, which is well known as LCM. With the help of the highest common factor, we can simplify and write the equations as simply as possible. Helps in algebra to solve equations easily. It 8s also used in real life while dealing with distribution, money, time, a delegation of work, etc. in many aspects.


3. Differentiate common factors and common multiples?

Ans. Before going to differentiate, let us recall about factors and multiples. Factors are the numbers that are exactly divided, whereas the multiples are the results from which we get after multiplying with factors. But both involve the process of multiplication. Let say,

 2*3= 6.

Here, 2,3 are the factors of 6

And, 6 is the multiple of 2,3.

In the same way, for 6 and 12, 2 and 3 are the common factors whereas, 6 is the common multiple for both 2 and 3.

Thus, common factors and common multiples have similarities as well as differences and both are helpful in solving complex equations.