Relation Between HCF And LCM

The words HCF and LCM, which stand for highest common factor and least common multiple, respectively, are used in this context. On the other hand, the HCF of any two or more numbers is the biggest factor that divides the number perfectly with no residue, whilst on the other hand, the LCM of any two or more two numbers is the smallest number that is divisible by the numbers in the specified order. In the case of the supplied positive integers “m” and “n,” the product of the HCF and LCM of these numbers is equal to the multiplication of the two numbers “m” and “n.” That is, HCF(m,n) × LCM (m,n) =m × n.

In the case of the supplied positive integers “m” and “n,” the product of the HCF and LCM of these numbers are equal to the multiplication of the two numbers “m” and “n.” That is, HCF(m,n) × LCM (m,n) =m × n.

Least Common Multiple(L.C.M)

When a set of numbers is divided into two groups of numbers, the Least Common Multiple (LCM) is defined as the lowest number that is a multiple of all of the numbers in each group.

Consider the following example: the LCM of 12 and 15 equals 60.

To get the LCM of a set of numbers, first list the multiples of each number that you want to find.

As a result, the multiples of 12 are equal to 12, 24, 36, 48, 60, 72, 84, and so on.

The multiples of 15 are as follows: 15, 30, 45, 60, 75, 90, 105, and so on.

Accordingly, 60 is the smallest number that is both 12 and 15 times larger than another number.

Highest Common Factor(H.C.F)

The Highest Common Factor (HCF) is defined as the largest number that divides evenly into all of the numbers from a group of numbers; it is also known as the greatest common factor (HCF).

For example, the HCF of the numbers 12 and 15 is three. Because the number 3 is the only common factor between the numbers 12 and 15, and it is the greatest number that divides both numbers, it is the most significant common factor.

Using the prime factorization of 12, we get: 2 x 2 x 3

Prime factorisation of 15 = 3 x 5

Relationship between HCF and LCM

The following shows the HCF-LCM relationship. Examine the relationship between HCF and LCM, and then use the relationships to and solve the problem in a straightforward manner.

In the case of the provided natural numbers, the LCM and HCF of the given numbers are equivalent to the product of the given numbers.

According to the stated property, LCM × HCF of a number = Product of the Numbers

Consider the following two numbers: A and B.

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 are as follows:

6 = 2 + 3 = 6

15 = 3 x 5

The LCM of 6 and 15 equals 30.

The HCF of 6 and 15 is equal to 3.

In the case of LCM (6, 15) and HCF (6, 15), the result is 30 x 3 = 90.

Product of 6 and 15 = 6 × 15 = 90

As a result, LCM (6, 15) ×  HCF (6, 15) equals Product(6, 15) = 90.

(ii)HCF AND LCM OF CO PRIME

 lcm of prime numbers(m, n) = product of two integers (m, n). Due to the fact that the HCF of co-prime numbers is equal to 1, the LCM of two co-prime numbers is the same as the product of the numbers in question. 

For example, 11 and 31 are two co-prime numbers. Let’s check to see if the LCM of the given co-prime numbers is equal to the product of the two numbers given above.

Solution: The factors 11 and 31 are as follows:

11 = 1 × 11

31 = 1 × 31

HCF of 11 and 31 = 1

LCM of 11 and 31 = 341

341 is the product of the numbers 11 and 31.

Recently, we proved to ourselves that the LCM of coprime numbers equals the product of the numbers.

iii) HCF and LCM of Fractions

When dealing with fractions such as m/n, p/q, u/v, and so on, we can use the following formula to get the HCF and LCM:

The LCM of fractions is equal to the LCM of Numerators ÷HCF of Denominators.

The HCF of fractions is equal to the HCF of Numerators ÷ LCM of Denominators.

Let’s look at two cases.

Example 1: Calculate the LCM of the following fractions: 1/4, 3/10, and 2/5

The LCM of fractions is equal to the LCM of Numerators ÷ HCF of Denominators.

LCM of fractions = LCM (1,3,2) ÷HCF(4,10,5) = 6 ÷ 1 = 6 LCM of fractions

Example 2: Calculate the high-frequency factor of the fractions 4/5, 5/2, and 6/7.

HCF of fractions = HCF of Numerators ÷ LCM of Denominators

HCF of fractions = HCF (4, 5, 6) ÷ LCM (5, 2, 7) = 1 / 70

CONCLUSION

The biggest factor existent between any two or more numbers is defined by the H.C.F., whereas the least common multiple (L.C.M.) is defined by the least common multiple that is exactly divisible by any two or more numbers. The greatest common factor (HCF) is also known as the greatest common factor (HCF), while the least common multiple (LCM) is also known as the least common multiple (LCM).

In any pair of numbers, the product of the HCF and LCM is always equal to the product of those two numbers, and the reverse is true as well. The same is not true, however, when there are three or more integers involved.