LCM and HCF
LCM Definition
In mathematics, the Least Common Multiple, abbreviated as LCM, is a fundamental concept. The LCM of two numbers, such as a and b, is represented as LCM(a, b). It is defined as the smallest positive integer that is divisible by both a and b.
For instance, consider the positive integers 4 and 6:
- The multiples of 4 are: 4, 8, 12, 16, 20, 24, and so on.
- The multiples of 6 are: 6, 12, 18, 24, and so forth.
The common multiples of 4 and 6 include: 12, 24, 36, 48, and so on. Among these, the smallest common multiple is 12. Hence, the LCM of 4 and 6 is 12.
Now, let's determine the LCM of 24 and 15:
- The multiples of 24 are: 24, 48, 72, 96, 120, and so on.
- The multiples of 15 are: 15, 30, 45, 60, 75, 90, 105, 120, and so forth.
The common multiples of 24 and 15 include: 120, 240, 360, etc. Among these, the smallest common multiple is 120. Therefore, the LCM of 24 and 15 is 120.
Also Check: Full Form of HCF
LCM of Two Numbers
Finding the LCM of 8 and 12
Let's try to find the Least Common Multiple (LCM) of two numbers, 8 and 12. We will look at the multiples of these two numbers to determine their LCM.
- The multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, and so on.
- The multiples of 12 are: 12, 24, 36, 48, 60, 72, 84, and so forth.
As you can see, the smallest common multiple of the numbers 8 and 12 is 24. Therefore, the LCM of 8 and 12 is 24.
Understanding HCF
In mathematics, HCF stands for Highest Common Factor. The largest positive integer that divides two or more positive integers without leaving a remainder is called the Greatest Common Divisor (GCD) or HCF.
Consider the numbers 8 and 12. The maximum number that can divide both 8 and 12 is 4. Therefore, the HCF of 8 and 12 is 4.
Also Chek: How to find HCF
Example: Finding the HCF of 24 and 36
To find the HCF of 24 and 36, we use prime factorization:
- 24 = 2 × 2 × 2 × 3
- 36 = 2 × 2 × 3 × 3
By factoring the numbers, we see that the common factors are 2 × 2 × 3. Therefore, the HCF of 24 and 36 is:
HCF(24, 36) = 2 × 2 × 3 = 12
HCF and LCM Formula
Combining HCF and LCM
The Highest Common Factor (HCF) and the Least Common Multiple (LCM) are related through a fundamental formula in mathematics. This relationship is given by the following formula:
A × B = HCF(A, B) × LCM(A, B)
Here, A and B are two numbers, HCF(A, B) is the Highest Common Factor of A and B, and LCM(A, B) is the Least Common Multiple of A and B.
Formulas in Terms of HCF and LCM
Using the relationship between HCF and LCM, we can derive the following formulas:
Finding HCF
The HCF of two numbers can be calculated as:
HCF of Two Numbers = (Product of Two Numbers) / (LCM of Two Numbers)
Finding LCM
The LCM of two numbers can be calculated as:
LCM of Two Numbers = (Product of Two Numbers) / (HCF of Two Numbers)
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Frequently Asked Questions on LCM and HCF
The rule for finding the Least Common Multiple (LCM) and Highest Common Factor (HCF) of two or more numbers is:
- Write each number as the product of its prime factors
- For LCM, multiply each prime factor by the highest power it occurs in any number
- For HCF, multiply the common prime factors with the lowest power they occur
The rule for finding the Highest Common Factor (HCF) of two or more numbers is:
- Write each number as the product of its prime factors
- List the common prime factors
- Multiply the common factors using the lowest power they occur
To find the Least Common Multiple (LCM) quickly, use the prime factorization method:
- Write each number as the product of its prime factors
- Multiply each prime factor by the highest power it occurs in any number
To find the Highest Common Factor (HCF) quickly, use the prime factorization method:
- Write each number as the product of its prime factors
- List the common prime factors
- Multiply the common factors using the lowest power they occur
The list method to find the Highest Common Factor (HCF) involves:
- List the factors of each number
- Find the common factors between the numbers
- The largest common factor is the HCF