Finding the Least Common Multiple (LCM) and Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of numbers is a fundamental concept in mathematics with applications in various fields. This guide provides a clear, step-by-step approach to calculating both LCM and HCF, catering to different levels of understanding.
Understanding LCM and HCF
Before diving into the calculations, let's clarify what LCM and HCF represent:
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Highest Common Factor (HCF) or Greatest Common Divisor (GCD): The largest number that divides exactly into two or more numbers without leaving a remainder. For example, the HCF of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18 evenly.
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Least Common Multiple (LCM): The smallest number that is a multiple of two or more numbers. For example, the LCM of 12 and 18 is 36 because 36 is the smallest number that is a multiple of both 12 and 18.
Method 1: Prime Factorization Method for LCM and HCF
This method is highly effective for finding both LCM and HCF, especially for smaller numbers.
Calculating HCF using Prime Factorization
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Find the prime factors of each number: Break down each number into its prime factors. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself (e.g., 2, 3, 5, 7, 11...).
Let's find the HCF of 12 and 18:
- 12 = 2 x 2 x 3 = 2² x 3
- 18 = 2 x 3 x 3 = 2 x 3²
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Identify common prime factors: Look for the prime factors that appear in both numbers.
In our example, both 12 and 18 have a '2' and a '3' as prime factors.
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Multiply the common prime factors: Multiply the common prime factors raised to the lowest power present in the factorizations.
The common prime factors are 2 and 3. The lowest power of 2 is 2¹ (from 12), and the lowest power of 3 is 3¹ (from 12). Therefore, HCF(12, 18) = 2 x 3 = 6
Calculating LCM using Prime Factorization
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Find the prime factors of each number: (Same as step 1 for HCF)
- 12 = 2² x 3
- 18 = 2 x 3²
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Identify all prime factors: List all the prime factors that appear in either number.
In our example, the prime factors are 2 and 3.
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Multiply the prime factors: Multiply each prime factor raised to the highest power present in the factorizations.
The highest power of 2 is 2² (from 12), and the highest power of 3 is 3² (from 18). Therefore, LCM(12, 18) = 2² x 3² = 4 x 9 = 36
Method 2: Division Method for HCF (Euclidean Algorithm)
This method is particularly efficient for finding the HCF of larger numbers.
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Divide the larger number by the smaller number: Find the remainder.
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Replace the larger number with the smaller number, and the smaller number with the remainder: Repeat step 1 until the remainder is 0.
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The last non-zero remainder is the HCF:
Let's find the HCF of 48 and 18:
- 48 ÷ 18 = 2 with a remainder of 12
- 18 ÷ 12 = 1 with a remainder of 6
- 12 ÷ 6 = 2 with a remainder of 0
Therefore, the HCF(48, 18) = 6
Method 3: Using the LCM and HCF Relationship
There's a useful relationship between the LCM and HCF of two numbers (a and b):
LCM(a, b) x HCF(a, b) = a x b
Once you've found either the LCM or HCF using the methods above, you can use this formula to calculate the other.
Practicing LCM and HCF
The best way to master LCM and HCF calculations is through practice. Try working through several examples using both methods. Start with smaller numbers and gradually progress to larger ones. You'll find that with practice, you'll become proficient in identifying LCM and HCF quickly and efficiently. Remember to choose the method that best suits the numbers you are working with. For smaller numbers, prime factorization is often easier, while for larger numbers, the Euclidean algorithm is more efficient for finding the HCF.
