HIGHEST COMMON FACTOR (HCF) GREATEST COMMON DIVISOR(GCD) OR GREATEST COMMON MEASURE (GCM)

Now before we understand the actual working of this Highest Common Factor, we need to get a clear understanding of what the highest common factor is. So in simple definition, the HCF is the largest number that can divide two or more positive numbers completely without leaving a remainder. The HCF is often studied with its counterpart the Least common multiple, however, we will be studying and focusing more on how we find the highest common factor. Once we have understood the working that is the how, we will learn its applicability the why, that is its applicability in our daily lives. 

WHAT IS THE HCF?

As previously mentioned it is the largest common number that can divide the two or more given numbers completely. The greatest common divisor is an apt name, but the highest common factor has been more used over time so we will be sticking to that. 

WORKING

Now the working of this is quite simple,  however, it could get confusing. 

I will be breaking the entire work of HCF into steps so it’s easier to understand. There are three key steps you need to remember using which one can find the HFC of any set of numbers.

1. List down both the individual numbers’ divisors, meaning numbers that will divide the given number completely leaving 0 as the remainder.
2. Compare both lists that divide each of the numbers completely
3. See which numbers lie in both the lists and whichever number is highest, it will be the HCF of that number.

So since the only way to learn or understand mathematical concepts is to practice and do sums pertaining to the mathematical concept. So let us get to finding the Highest common factor for a few numbers.

 Example 1) what is the HCF of 8 and 24.

Solution–

Step 1

Numbers that are a factor of 8 or divided by 8 completely

1, 2,4, 8. 

Numbers that are a factor of 24 or divided by 24 completely

1,2,4,6,8,12,24.

Step 2

 On comparing the list, the common factors are

1,2,4,8.

Step 3

The highest number on the list is 8. So therefore 8 is the Highest common factor for 8 and 24 is 8 itself. 

Example 2)

 Let’s find the Highest common factor of the numbers 12, 36 and 100

solution

Step 1 – factors of 12

1,2,3,4,6,12

Factors of 36

1,2,3,4,6,12,18,36

Factors of 100

1,2,4,5,10.20,25,50,100.

Step 2

The common factors that divide all three are

1,2,4

Step 3

Since the number 4 is the highest in the lot, it is the HCF for 12,36 and 100.

WHAT DO WE DO WITH THE HCF?

The highest common factor has no direct applicability in our daily lives. However, deeming it useless will be grossly incorrect. HCF is heavily used in algebra. It enables one to find the relationships and nature of relationships between various polynomials and their correlation in many fields.

WHAT IS LCM?

As previously mentioned the HCF is often studied with the Least common multiple. LCM is in some way opposite to the HCF. The HCF is always gonna be smaller than the number, while the LCM is always gonna be greater than the number.

To better understand the LCM and its difference from HCF, we need to get a clearer understanding of both the difference between a factor and a multiple is.

A factor, ‘x’, is one that will divide the numbers A and B completely.

A multiple ‘y’ is one that will be divided by the numbers themselves.

For example

Let the numbers be 4 and 8

 Now the factors of the numbers will be 1,2,4.

The Multiple of the numbers will be 8,16,24,32 and so on.

 Since we have already learnt how we find the highest common factor, I will skip the steps.

In the above, the HCF is 4, while the Least common multiple will be 8. 

CONCLUSION

Now that the question as to how we find the highest common factor has been answered, it is gonna be slightly easier to understand algebraic correlations. Just to run through things again, HFC is nothing but the highest number that divides the set of given numbers. LCM however is different as it is the lowest number that can be divided by the given set of numbers. The HCF is always gonna be lower than the LCM for the same set of given numbers. These two concepts are often studied together, however, we stuck to HCF.