Easy way to solve HCF (GCD)

Solving HCF easily through Observation

HCF or the Highest common factor is nothing but the highest factor of the two given numbers which can divide the given numbers. In our last blog we had shown the easy way of finding the LCM ( lowest common multiple) just by inspecting. Today along with the factorization  and Venn diagram method we are also going to discuss a method which helps in finding the solutions easily specially when you are trying to solve these questions in the competitive exams.

Remember in the least common multiple we have to find the number that can be divided by the numbers given for which we need to find the LCM. However in case of HCF the it is just opposite I,e we  need to find the number that will divide the given numbers for which the HCF has to be found.

For example in case we need to find the HCF of 2 and 5 then the factors for 2 are 1 and 2  & factors of 5 are 1 and 5 . The factor that can divide both the numbers js 1 hence the highest common denominator or factor is 1.

Note : For the prime numbers the highest common factor or the greatest common denominator is 1 always

 

 Lets see the factorization method for finding the HCF for 24 and 36

The factors of 24  and 36 can be shown as below

 

From the above we can see that 2x2x3 is the factor repeated in both and can divide both 24 and 36 so this is the greatest common denominator or highest common factor.

The same can be represented by the Venn diagram as shown below

As we can see the common factors are taken in the   middle and are the Highest Common factors

 

Another way of finding the HCF (GCD) is the repeated division method. This is what we normally teach in schools. The repeated division method is as shown below.

However, we can find the HCF (GCD) without resorting to any of these methods. Lets see how this can be done. For example, if we have to find the HCF of lets say 8 and 12.

Step 1 : Find the difference between the given numbers. So 12-8 = 4

Step 2: Check whether this divides both the given numbers.

In this case it does so our Highest Common Factor is

4 . If you want you can check this using any of the above-given methods

Answer :  HCF is 4.

Now lets take another pair of numbers. Lets take 21 and 35.

 

Step1: The difference between the two numbers is 35-21 = 14

Step2 : Check whether 14 divides both the numbers. In this case it doesn’t.

Step3: Find the factors for number 14 which is 2x7.

Step 4: Check which of the numbers will divide both 21 and 35. In this case we find that 7 divides both 21 and 35.

Answer is HCF = 7

Now lets see if there are 3 numbers and the HCF needs to be found what should be done.

For example lets take 12, 40  and 86

We can take the difference between any two of these numbers

40-12 = 28, 86-40 = 46 , 86-12=74

Now lets find the factors for 28 which will be 7x2x2

Factors of 46 are 23x2

Factors of 74 are 37x2

The number that divides all the numbers is 2 so HCF is 2

Just observation can give you the solution of some problems that you found difficult