HCF of Two Numbers
HCF of two numbers is the highest common number, which is available in both the numbers. Before we proceed ahead to find the HCF, let us discuss what HCF is. HCF or highest common factor is the factor of any two or more numbers, which are common among them. Sometimes, it is also called the greatest common factor (GCF) or greatest common divisor (GCD).
For example, the HCF of 2 and 4 is 2, because 2 is the number which is common to both 2 and 4. For such small numbers, finding HCF is an easy method. But for larger numbers, we need to use different techniques such as prime factorisation and long division method, to find the HCF.
Let us discuss both the techniques.
HCF of two numbers By Prime Factorisation
Let us solve some examples here to understand this method.
Question 1: What is the HCF of 24 and 36?
Solution: By prime factorisation, we can write the two numbers;
24 = 2 x 2 x 2 x 3
36 = 2 x 2 x 3 x 3
Hence, after factoring the numbers 24 and 36, we can see, the factors 2x2x3 are common.
Therefore, the HCF (24, 36) = 2x2x3 = 12
Question 2: What is the HCF of 35 and 55?
Solution: By prime factorisaton we can write the two numbers as:
35 = 5 x 7
55 = 5 x 11
Hence, we can see the highest common factor for 35 and 55 here is 5.
Therefore, HCF (35, 55) = 5
HCF of two numbers by Division Method
We have already understood the prime factorisation method to determine the HCF.
Steps for Division method:
If we were given two numbers, then
- First, divide the large number by a small number.
- If the remainder is left, then divide the first divisor by remainder.
- If the remainder divides the first divisor completely, then it is the HCF or highest common factor of the given two numbers.
- If the remainder does not divide the first divisor completely, then repeat the steps.
Let us now learn the division method with the help of examples
Question:1 What is the HCF of 120 and 100.
Solution: Now, let us find the HCF by using division method.
Divide 120 by 100.
120/100 → 1 and remainder is 20
Now, divide the first divisor 100 by first remainder 20.
100/20 → 5 and remainder is 0.
Therefore, 20 is the HCF of 120 and 100.
Question 2: Find the HCF of 45 and 60 by the division method.
Solution: Divide 60 by 45.
60/45 → 1 and remainder is 15
Now, divide 45 by 15
45/15 → 3
Therefore, 15 is the HCF of 45 and 60.
HCF of Three Numbers
Let us solve an example when we need to find the HCF of three numbers.
Example: Find the HCF of 126, 162 and 180.
Solution: By prime factorisation, we can write the given numbers as;
126 = 2x3x3x7
162 = 2x3x3x3x3
180 = 2x2x3x3x5
Taking out the common factors of 126, 162 an d180, we get:
HCF(126,162,180) = 2x3x3 = 18