Relation between HCF and LCM
HCF and LCM are the two terms that stand for highest common factor and least common multiple respectively. The HCF is the greatest factor of two numbers or more than two numbers which divides the number exactly with no remainder, while on the contrary the LCM of two numbers or more than two numbers is the smallest number which is divisible by given numbers exactly. After discussing the definition of HCF and LCM, in this article, we will focus on the relation between HCF and LCM along with solved examples and practice questions.
HCF and LCM Relation
The multiplication of common prime factors of given numbers with the least exponential powers is the HCF of two numbers or more numbers. For example, The HCF of 12 and 20 is 4.
Prime factors of 12 = 2 × 2× 3
Prime factors of 20 = 2 × 2 ×5
HCF is 2 × 2=4
The LCM or the least common multiple is the smallest natural number which is a multiple of two or more numbers. For example, the LCM of 12 and 20 is 60.
Multiples of 12 = 12, 24, 36, 48, 60, 72, 84 and so on.
Multiples of 20 = 20, 40, 60, 80 and so on.
LCM of 12 and 20 is 60.
HCF and LCM of two numbers are related to each other as well as with the given numbers. For any two numbers a and b, HCF (a, b) × LCM (a, b) = a × b.
HCF and LCM of Positve Integers
The product of HCF and LCM of the given positive integers suppose “m” and “n” is equal to the multiplication of the given numbers “m” and “n”. That is, HCF(m,n) × LCM (m,n) =m × n. Let us look at the example based on the above relation.
Example: Prove that the LCM (8, 12) × HCF (8, 12) = Product(8, 12)
Solution: LCM and HCF of 8 and 12:
Multiples of 8 and 12 to find LCM is 8 = 8, 16, 24 , 32, 40 and so on, and 12 = 12, 24, 36, 48 and so on. LCM of 8 and 12 = 24. Now let us look at factors of 8 and 12.
8 = 2 × 2 × 2
12 = 2 × 2 × 3
HCF of 8 and 12 = 4.
LCM (8, 12) × HCF (8, 12) = 24 × 4 = 96
Product of 8 and 12 = 8 × 12 = 96
Hence, LCM (8, 12) × HCF (8, 12) = Product(8, 12) = 96
HCF and LCM of Co-Prime numbers
LCM of Co-Prime numbers(m, n) = Product of two numbers (m, n). Since 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. Look at the given example to verify the relation.
For example: 11 and 31 are two co-prime numbers. Let's verify LCM of given co-prime Numbers is equal to the product of the given numbers.
Solution: Factors of 11 and 31 are,
11 = 1 × 11
31 = 1 × 31
HCF of 11 and 31 = 1
LCM of 11 and 31 = 341
Product of 11 and 31 = 11× 31 = 341
We just verified that the LCM of co-prime numbers = Product of the numbers
HCF and LCM of Fractions
To find HCF and LCM of fractions like m/n, p/q, u/v, etc, we can use the below-mentioned formula:
LCM of fractions = LCM of Numerators ÷ HCF of Denominators
HCF of fractions = HCF of Numerators ÷ LCM of Denominators
Let's take two examples to understand it better.
Example 1: Find the LCM of the given fractions 1/4, 3/10, 2/5.
LCM of fractions = LCM of Numerators ÷ HCF of Denominators
LCM of fractions = LCM (1,3,2) ÷ HCF(4,10,5) = 6 ÷ 1 = 6
Example 2: Find the HCF of the fractions 4/5, 5/2, 6/7.
HCF of fractions = HCF of Numerators ÷ LCM of Denominators
HCF of fractions = HCF (4, 5, 6) ÷ LCM (5, 2, 7) = 1 / 70
Given below is the image to summarize the relation between HCF and LCM.
Relation between HCF and LCM of 3 Numbers
In this section, we are going to learn the relation between HCF and LCM of three numbers. Suppose p,q,r are the three numbers, then to find LCM of these three numbers we will multiply the product of numbers (p×q×r) with HCF of numbers(p,q,r) and divide it by the product of HCF(p,q), HCF(q,r) and HCF(r,p). Similarly to find the HCF of p, q, r we will multiply the product of numbers (p×q×r) with LCM of numbers(p,q,r) and divide it by the product of LCM(p,q), LCM(q,r) and LCM(r,p). The below-mentioned formulae can be used to understand the relation and to calculate the HCF and LCM of 3 numbers.
- \(LCM (p, q, r) = \dfrac{(p\times q\times r )\times HCF (p, q, r)}{HCF(p,q)\times HCF(q,r)\times HCF(r,p)}\)
- \(HCF (p, q, r) = \dfrac{(p\times q\times r) \times LCM(p, q, r)}{LCM(p,q)\times LCM(q,r)\times LCM(r,p)}\)
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FAQs on Relation Between HCF and LCM
What is the Relationship Between HCF and LCM of Two Numbers?
The relationship between HCF and LCM of two given positive integers, let's say “m” and “n” is equal to the multiplication of the given numbers “m” and “n”, given as, HCF(m,n) × LCM (m,n) =m × n
What is LCM of Two Numbers?
The LCM or the least common multiple is the smallest natural number which is a multiple of two or more numbers. For two numbers, we can find the LCM by listing down all the multiples and then selecting the least common multiple of both. Another method is using the prime factorization method.
What is the Highest Common Factor?
The multiplication of common prime factors of given numbers with the least exponential powers is the highest common factor of two or more numbers.
What is the Relation Between HCF and LCM of co-prime numbers?
The Relation Between HCF and LCM of co-prime numbers is the product of the numbers = LCM of Co-Prime numbers, as the HCF of co-prime numbers is equal to 1.
What are H.C.F. and L.C.M. of Fractions?
The HCF and LCM of fractions is given as,
- LCM of fractions = LCM of Numerators ÷ HCF of Denominators
- HCF of fractions = HCF of Numerators ÷ LCM of Denominators
What is the Difference Between LCM and HCF?
The LCM of two or more numbers can be divided by the numbers and it is the smallest common multiple of those numbers. The greatest number that divdes the two or more numbers exactly is the highest common factor of those numbers. HCF of two numbers cannot be greater than the numbers, while LCM of two numbers is always greater than or equals to the larger number except 0 which is considered as the common multiple of every number.