HIGHEST COMMON FACTOR

The greatest common factor of two or more whole numbers is the largest whole number that divides evenly into each of the numbers. There are two ways to find the greatest common factor.
The first method is to list all of the factors of each number, then list the common factors and choose the largest one.
 Find the GCF of 36 and 54.
The factors of 36 are:1, 2, 3, 4, 6, 9, 12, 18, and 36.
The factors of 54 are:1, 2, 3, 6, 9, 18, 27, and 54.
The common factors of 36 and 54 are:1, 2, 3, 6, 9, 18
Although the numbers in bold are all common factors of both 36 and 54, 18 is the greatest common factor.

Steps to find Hcf:

 Step1: Find prime factorisation of the given numbers.

Step2: Take the product of the common prime numbers with lowest power(which is common in all the given numbers)

Let’s use the same numbers, 36 and 54 again to find their greatest common factor.
Step1:The prime factorization of 36 is:2 x 2 x 3 x 3=2²x3²
The prime factorization of 54 is:2 x 3 x 3 x 3=2¹x3³
Notice that the prime factorizations of 36 and 54 both have one 2 and two 3s in common. So, we simply multiply these common prime factors to find the greatest common factor. Like this…
2¹x3²= 18.


Find the highest common factor of 900 and 270 .
 Express 900 as a product of its prime factors: 2²x3²x5²
Express 270 as a product of its prime factors: 2¹x3³x5¹
The HCF of 900 and 270 is: 2¹x3²x5¹=90( multiplying together the factors which appear in both lists above)


Find the HCF of 16 and 40 .
Prime factorisation of 16 = 2 x 2 x 2 x 2=2⁴
Prime factorisation of 40 = 2 x 2 x 2 x 5=2³x5¹
Common prime factors = 2³
HCF of 16 and 40 = 2³ = 8

Note:

HCF of fractions is  ratio of hcf of numerators to lcm of denominators

 Hcf of given numbers is always less than or equal to the least number in the given numbers.
If there is no common prime number in the given numbers then 1 is the hcf of the given numbers.
Hcf is always a factor of lcm.
Hcf is always a factor of each number in the given numbers.
Product of two numbers is equal to lcm x hcf.
If hcf of two numbers is 1, then the are said to be co-primes; (4,17), (13,19), (8,9) are examples of co-prime
PRACTISE-1