H.C.F and L.C.M

The full form of LCM in Maths is the Least Common Multiple, whereas the full form of HCF is the Highest Common Factor. The H.C.F. defines the greatest factor present in between given two or more numbers, whereas L.C.M. defines the least number which is exactly divisible by two or more numbers. H.C.F. is also called the greatest common factor (GCF) and LCM is also called the Least Common Divisor.

To find H.C.F. and L.C.M., we have two important methods which are the Prime factorization method and the division method. The shortcut method to find both H.C.F. and L.C.M. is a division method.

LCM (Least Common Multiple)

In arithmetic, the least common multiple or LCM of two numbers say a and b, is denoted as LCM (a,b). And the LCM is the smallest or least positive integer that is divisible by both a and b.

For example, let us take two positive integers 4 and 6.

Multiples of 4 are: 4,8,12,16,20,24…

Multiples of 6 are: 6,12,18,24….

The common multiples for 4 and 6 are 12,24,36,48…and so on. The least common multiple in that lot would be 12.

Let us find the LCM of 15,30,90.

HCF and LCM

LCM of (15,30,90) =  2×3×5×3 = 90.

Highest Common Factor (HCF)

The HCF or Highest Common Factor of two or more numbers is the greatest common factor of the given set of numbers. In other words, HCF is the greatest number which exactly divides two or more given numbers.

HCF by Listing Method

The listing method involves the process of listing the factors of the given numbers.

For example, find the HCF of 20 and 35.

  • All possible factors of 20 are 1,2,4,5,10 and 20
  • All possible factors of 60 are 1,3,4,5,6,10,12,15,20,30,60

The common factors of the given numbers are : 1,2,4,5,10,20. The greatest among all other numbers is 20, so it shall be the HCF of both the numbers.

HCF by Prime Factorization

Before finding HCF by prime factorization we need to know the concept of the same. Let’s take a number say, 45. Now the factors of 45 are 1,3,5,9,15 and 45 itself. Now, apart from 3 and 5 the other numbers 9 and 15 are composite numbers. We hence further factorize them with 9= 3×3 and 15=3×5.

So the factors of 45 shall be only 1,3,3, and 5. This is prime factorization. We now define prime factorization as the process of expressing the number as the product of its prime factors. The prime factors include only prime numbers and not composite numbers.

When we find HCF by prime factorization method, we are finding the greatest common factor among the prime factors or numbers.

Find the HCF of  36 and 48.

Finding prime factors individually:

HCF and LCM

  • All possible factors of 36 are: 2×2×3×3×1
  • All possible factors of 48 are: 2×2×2×2×3×1

Choose out the common factors: 2×2×3

Multiply all the common factors to get the HCF of the given numbers

Here the given numbers are 36 and 48. The product of the common factors: 2×2×3 = 12. So the HCF for the numbers 36 and 48 is 12.