Relation Between HCF and LCM

To learn the relation between H.C.F. and L.C.M. of two numbers or the given n numbers first we need to know about the definition of the Highest Common Factor (H.C.F.) and the Least Common Multiple (L.C.M) and also LCM and HCF formulas. In this article, we are going to discuss the definition and the relation between HCF and LCM of given numbers in detail with examples.

Least Common Multiple(L.C.M)

The Least Common Multiple (LCM) is defined as the smallest number that is a multiple of all the numbers from a group of numbers.

Consider an example, the LCM of 12 and 15 is 60.

To find the LCM of numbers, first, mention the multiples of each number.

Therefore, the multiples of 12 = 12, 24, 36, 48, 60, 72, 84.. etc.,

The multiples of 15 = 15, 30, 45, 60, 75, 90, 105,.. Etc,

So, 60 is the smallest number that is a multiple of both 12 and 15

Highest Common Factor(H.C.F)

The Highest Common Factor ( HCF) is defined as the largest number that divides evenly into all the numbers from a group of numbers.

For example, the HCF of 12 and 15 is 3. Because 3 is the only common factor for both the numbers 12 and 15 and it is the largest number that divides both the numbers.

Prime factorisation of 12 = 2 x 2 x 3

Prime factorisation of 15 = 3 x 5

HCF and  LCM Relation

The followings are the relation between HCF and LCM. Go through the relation between HCF and LCM, and solve the problem using the relations in an easy way.

(i) The product of LCM and HCF of the given natural numbers is equivalent to the product of the given numbers.

From the given property, LCM × HCF of a number = Product of the Numbers

Consider two numbers A and B, then.

Therefore,LCM (A , B) × HCF (A , B) = A × B

Example 1: Show that the LCM (6, 15) × HCF (6, 15) = Product(6, 15)

Solution: LCM and HCF of 6 and 15:

6 = 2 × 3

15 = 3 x 5

LCM of 6 and 15 = 30

HCF of 6 and 15 = 3

LCM (6, 15) × HCF (6, 15) = 30 × 3 = 90

Product of 6 and 15 = 6 × 15 = 90

Hence, LCM (6, 15) × HCF (6, 15)=Product(6, 15) = 90

(ii) The LCM of given co-prime numbers is equal to the product of the numbers since the HCF of co-prime numbers is 1.

So, LCM of Co-prime Numbers = Product Of The Numbers

Example 2: 17 and 23 are two co-prime numbers. By using the given numbers verify that,

LCM of given co-prime Numbers = Product of the given Numbers

Solution: LCM and HCF of 17 and 23:

17 = 1 x 7

23 = 1 x 23

LCM of 17 and 23 = 391

HCF of 17 and 23 = 1

Product of 17 and 23 = 17 × 23 = 391

Hence, LCM of co-prime numbers = Product of the numbers

(iii) H.C.F. and L.C.M. of Fractions

LCM of fractions = LCM of Numerators / HCF of Denominators

HCF of fractions = HCF of Numerators / LCM of Denominators

Example 3: Find the LCM of the fractions 1 / 2 , 3 / 8, 3 / 4

Solution:

LCM of fractions = LCM of Numerators/HCF of Denominators

LCM of fractions = LCM (1,3,3)/HCF(2,8,4)=3/2

Example 4: Find the HCF of the fractions 3 / 5, 6 / 11, 9 / 20

HCF of fractions HCF of Numerators/LCM of Denominators

HCF of fractions = HCF (3,6,9)/LCM (5,11,20)=3/220

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