Lowest Common Multiple (LCM) and the Highest Common Factor (HCF), are closely connected. They play a key role in simplifying mathematical statements, prime factorization, and finding relationships between numbers, especially when dealing with fractions. To understand the relationship between the HCF and LCM of two or more numbers, one must grasp the concepts of LCM and HCF and know how to apply the relevant formulas.
HCF and LCM Relation
This article provides a detailed explanation, with examples, to clarify the connection between the HCF and LCM values given.
LCM (Lowest Common Multiple) Definition
Least Common Multiple (LCM) is the smallest number, a multiple of every number in a set. For example, the LCM of 12 and 15 is 60, the smallest number divisible by both 12 and 15.
To find the LCM, list the multiples of each number and determine the lowest common multiple. In this case, 60 is the LCM of 12 and 15, the smallest number that is a multiple of both.
HCF (Highest Common Factor) Definition
Highest Common Factor (HCF) is the greatest integer that divides two or more numbers without a residual. For example, the HCF of 12 and 18 is 6 because 6 divides both numbers evenly. HCF is also known as the greatest common divisor (GCD). It is the highest number that evenly divides all numbers in a given set.
Example: HCF of 12 and 15 is 3
- Prime factorization of 12 = 2 × 2 × 3
- Prime factorization of 15 = 3 × 5
HCF = 3
Relation between HCF and LCM
Relation between HCF and LCM is stated below:
Product of two numbers is equal to product of their LCM and HCF, i.e.
(HCF of Two Numbers) × (LCM of Two Numbers) = Product of two Numbers
For example: 10 and 11 are coprime numbers.
- 10 = 1 × 1
- 11 = 1 × 11
LCM of 10 and 11 = 110
HCF of 10 and 11 = 1
Product of 10 and 11 = 10 × 11 = 110
Consider two numbers A and B, then
Therefore, LCM (A , B) × HCF (A , B) = A × B
Special Cases Of HCF And LCM
When dealing with rational numbers, understanding the rules and ratios of HCF and LCM is crucial. In cases where more than two numbers are involved, the process remains the same.
By multiplying the numbers and their HCF, and dividing them by their respective pairs’ HCF, we can calculate the LCM.
The same applies when finding the HCF, using the LCM in place of the HCF in the formula.
Consequently,
LCM = (x×y×z) × (HCF of x, y, z)/ HCF (x, y) × HCF (y, z) × HCF (x, z)
To find the HCF, the inverse formula needs to be used.
HCF (x,y and z) = (x×y×z) × (LCM of x, y, z)/ LCM (x, y) × LCM (y, z) × LCM (x, z)
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Examples on on Relation Between HCF and LCM
Example 1: Show that the LCM (6, 15) × HCF (6, 15) = Product of (6, 15)
Solution:
LCM and HCF of 6 and 15
6 = 2 × 3
15 = 3 × 5
LCM of 6 and 15 = 30
HCF of 6 and 15 = 3
LCM (6, 15) × HCF (6, 15) = 30 × 3 = 90…(i)
Product of Two Numbers = 6 × 15 = 90…(ii)
Thus, LCM (6, 15) × HCF (6, 15) = Product of (6, 15)
Example 2: Show that the LCM (8, 12) × HCF (8, 12) = Product of (8, 12)
Solution:
LCM and HCF of 8 and 12
8 = 2 × 2 × 2
12 = 2 × 2 × 3
LCM of 8 and 12 = 2 × 2 × 2 × 3 = 24
HCF of 8 and 12 = 2 × 2 = 4
LCM (8, 12) × HCF (8, 12) = 24 × 4 = 96…(i)
Product of Two Numbers = 8 × 12 = 96…(ii)
Thus, LCM (8, 12) × HCF (8, 12) = Product of (8, 12)
Example 3: Show that the LCM (9, 10) × HCF (9, 10) = Product of (9, 10)
Solution:
LCM and HCF of 6 and 15
9 = 3 × 3
10 = 2 × 5
LCM of 9 and 10 = 2 × 3 × 3 × 5 = 90
HCF of 9 and 10 = 1
LCM (6, 15) × HCF (6, 15) = 90 × 1 = 90…(i)
Product of Two Numbers = 9 × 10 = 90…(ii)
Thus, LCM (9, 10) × HCF (9, 10) = Product of (9, 10)
FAQs on Relation Between HCF and LCM
If LCM and HCF of two numbers are 3 and 2 respectively, and one of the numbers is 6 then another number is?
We know that, LCM × HCF = a × b where a and b are two numbers.
3 × 2 = 6 × b
b = 1
What is the significance of finding the HCF and LCM of numbers?
HCF and LCM are crucial for mathematical procedures like simplification, common denominator finding, equation solving, and pattern recognition.
What is the relationship between the HCF and LCM of three numbers?
For three numbers a, b, and c, the relationship between their HCF and LCM can be expressed as:
HCF(a, b, c) × LCM(a, b, c) = Product of Three Numbers
Can the HCF and LCM of two numbers be the same?
It is possible for two numbers to have the same HCF and LCM if they are equal. In these circumstances, the number itself would be the HCF and LCM.