Hello All, Here we are going to discuss the Hightes Common Factor or HCF. Before starting our tricks to find the HCF of a number. Let us know what is HCF.
What is HCF?
“HCF” stands for highest common factor (A common factor can be defined as a number that divides the other number evenly with no remainder left).
It can be defined as the method to find out the highest common factor between two numbers or more. “HCF” is also called “GCM” which stands for greatest common measure and “GCD” which stands for the greatest common divisor.
Let’s understand the definition in more detail of HCF with the help of an example.
We can take another example for better clarity of the concept:-
Example: Find HCF of 50 and 75 will be:-
Answer: Method 2:
- 50 = 2 x 5 x 5
- 75 = 3 x 5 x 5
The product of all Common Prime Factors is called HCF. Here, the common factor is 5 with the highest power 2.
So, the HCF will be 5*5 = 25
Answer: Method 1:
We can perform our HCF with our general method too, which is described previously.
Again, our HCF is 25. It means 25 is the greatest number or divisor which can divide both 50 and 75.
Answer: Method 3:
Now, we can justify our HCM too, let’s consider the factors of 50 and 75, we get;
- 50 → 5, 10, 25, 50…
- 75 → 5, 15, 25, 75…
Here, we can see that the highest common factor for 50 and 75 is 25. This justifies the method of HCF as correct.
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Properties of HCF
- The HCF of prime numbers cannot be other than 1 or we can say that it is always equal to 1.
For example: – HCF of (5, 7) is 1 as both are prime numbers.
- The Product of LCM *HCF is equal to the product of numbers itself (This property is only applicable to two numbers).
For example : – LCM (15, 20) = 60
HCF (15,20) = 5
LCM*HCF = 60*5 = 300
Product of numbers = 15* 20 = 300
Hence, the product of LCM and HCF is equal to the product of numbers.
- The HCF of given numbers cannot be greater than the numbers themselves.
For example: – HCF of (5, 7) cannot be greater than 5 and 7. It can only be less than 7.
How to Find the HCF of Two and Three Numbers
For smaller numbers, you can simply write the factors of both numbers and can spot the greatest common factor.
Example:- 6 and 8
- 6 = 2, 3, 6,……
- 8 = 2, 4, 8……
Here, 2 is the HCF of (6, 8).
We can find our HCF by 2 methods that are: –
- By Prime Factorization
- By Division Method
1. By Prime Factorization
Example: Suppose, we have 3 numbers 12, 18 and 24. We have to write the prime factors of each number individually.
- 12 = 2x 2 x 3
- 18 = 2 x 3 x 3
- 24 = 2 x 2 x 2 x 3
Now, write the prime factors of all numbers together: –
12, 18, 24 = 2 x 2 x 3 x 2 3 x 3 x 2 x 2 x 2 x 3
Here, the common factors are 2 and 3 with the highest power 1.
So, the HCF will be 2 x 3 = 6
HCF of (12, 18, 24) is 6.
2. By Division Method
Steps:
- First, write all the numbers, and separate them by commas
- Now, divide the numbers by the smallest prime number.
- Write down the quotients.
- We have to keep on repeating the process until there is no coprime number left.
- Now, HCF of the numbers will be equal to the product of all the prime numbers we obtained in the method.
Let us understand the division method with the help of an example: –
Example 1. Find HCF of 36 and 24 by the division method?
Let’s draw a table and find out according to the previous steps:
| 1ST NUMBER | 2ND NUMBER | |
| 2 | 36 | 24 |
| 2 | 18 | 12 |
| 3 | 9 | 6 |
| 3 | 2 | |
Therefore, HCF of (36, 24) = 2 x 2 x 3 = 12
Example 2. Find HCF of 36, 24 and 48 by the division method?
Let’s draw a table and find out the HCF:
| IST NUMBER | 2ND NUMBER | 3RD NUMBER | |
| 2 | 36 | 24 | 48 |
| 2 | 18 | 12 | 24 |
| 3 | 9 | 6 | 12 |
| 3 | 2 | 4 |
Therefore, HCF of (36, 24, 48) = 2 x 2 x 3 = 12
Shortcut Method to find out HCF in 10 SECONDS.
This is the general way to find the HCF and it is the simplest.
Step 1 : – Firstly, Divide the larger number by the smaller number. such as: – Larger Number/Smaller Number
Step 2 : – Now, Divide the divisor obtained from step 1 by the remainder left.
Such as : – Divisor of step 1/Remainder
Step 3 : – Then, Again divide the divisor obtained from step 2 by the remainder.
Such as : – Divisor of step 2/Remainder
Step 4 : – Keep on repeating the process until the remainder comes to zero.
Step 5 : – The divisor obtained from the last step will be the HCF.
Now, let’s solve an Example : – Find HCF of 5 and 13
Hence, HCF of 5 and 13 is 1.
HCF Of Three Numbers
Step 1 : – First, calculate the HCF of any 2 numbers.
Step 2 : – Now, find the HCF of the third number with the HCF obtained from step 1.
Step 3 : – The HCF obtained from step 2 will be the final HCF of the three numbers.
Now, let’s Solve an Example : – Find Hcf of 9, 27, 30
Firstly, calculate the HCF of any two numbers.
Here, i’m taking 9 and 27
PRACTICE (By Shortcut Method)
- 15 and 25
- 30, 45 and 60
- 48, 56 and 60
- 10 and 15
- 43, 91and 183
PRACTICE (By Prime Factorisation Method)
- 72 and 102
- 102 and 74
- 56,78 and 93
- 46 and 69
- 45, 99 and 162
PRACTICE (By Division Method)
- 6, 8 and 18
- 55 and 90
- 34 and 68
- 78, 90 and 110
- 14, 20 and 46
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