Hcf Of 120 And 150

Finding the Highest Common Factor (HCF) of 120 and 150: A Comprehensive Guide

Finding the highest common factor (HCF), also known as the greatest common divisor (GCD), of two numbers is a fundamental concept in mathematics with applications ranging from simplifying fractions to solving complex algebraic problems. This article provides a comprehensive guide to determining the HCF of 120 and 150, exploring various methods and delving into the underlying mathematical principles. We'll cover different approaches, from prime factorization to the Euclidean algorithm, ensuring a thorough understanding for learners of all levels.

Understanding Highest Common Factor (HCF)

The HCF of two or more numbers is the largest number that divides each of them without leaving a remainder. In simpler terms, it's the biggest number that's a factor of both numbers. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The common factors of 12 and 18 are 1, 2, 3, and 6. The highest of these common factors is 6, therefore, the HCF of 12 and 18 is 6.

Understanding the HCF is crucial for simplifying fractions. For example, the fraction 12/18 can be simplified to 2/3 by dividing both the numerator and denominator by their HCF, which is 6. This simplifies calculations and makes working with fractions easier.

Method 1: Prime Factorization

This method involves breaking down each number into its prime factors – numbers divisible only by 1 and themselves. Once we have the prime factorization of both numbers, we identify the common prime factors and multiply them to find the HCF.

Let's apply this to find the HCF of 120 and 150:

1. Prime Factorization of 120:

• 120 = 2 x 60
• 60 = 2 x 30
• 30 = 2 x 15
• 15 = 3 x 5

Therefore, the prime factorization of 120 is 2³ x 3 x 5.

2. Prime Factorization of 150:

• 150 = 2 x 75
• 75 = 3 x 25
• 25 = 5 x 5

Therefore, the prime factorization of 150 is 2 x 3 x 5².

3. Identifying Common Prime Factors:

Both 120 and 150 share the prime factors 2, 3, and 5.

4. Calculating the HCF:

The lowest power of the common prime factors is: 2¹, 3¹, and 5¹. Multiplying these together: 2 x 3 x 5 = 30

Therefore, the HCF of 120 and 150 is 30.

Method 2: Listing Factors

This method is suitable for smaller numbers. We list all the factors of each number and then identify the largest common factor.

1. Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

2. Factors of 150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150

3. Common Factors: 1, 2, 3, 5, 6, 10, 15, 30

4. Highest Common Factor: The largest number in the list of common factors is 30.

Therefore, the HCF of 120 and 150 is 30. This method, while straightforward, becomes less efficient with larger numbers.

Method 3: Euclidean Algorithm

The Euclidean algorithm is a highly efficient method for finding the HCF of two numbers, especially large ones. It's based on the principle that the HCF of two numbers does not change if the larger number is replaced by its difference with the smaller number. This process is repeated until the two numbers are equal, and that number is the HCF.

Let's apply the Euclidean algorithm to find the HCF of 120 and 150:

1. Start with the larger number (150) and the smaller number (120): 150, 120

2. Divide the larger number by the smaller number and find the remainder: 150 ÷ 120 = 1 with a remainder of 30.

3. Replace the larger number with the smaller number (120) and the smaller number with the remainder (30): 120, 30

4. Repeat the division: 120 ÷ 30 = 4 with a remainder of 0.

5. Since the remainder is 0, the HCF is the last non-zero remainder, which is 30.

Therefore, the HCF of 120 and 150 is 30. The Euclidean algorithm is significantly more efficient than the listing factors method for larger numbers, making it a preferred choice for computational purposes.

Mathematical Explanation: Why These Methods Work

The success of all three methods hinges on fundamental properties of divisors and factors. The prime factorization method directly reveals the common building blocks (prime factors) of the numbers, allowing us to construct the largest possible common factor. The listing factors method is a brute-force approach, systematically identifying all possibilities until the largest common factor is found. The Euclidean algorithm, however, utilizes a more elegant approach by exploiting the relationship between the HCF and the remainders in the division process. Each step in the Euclidean algorithm maintains the invariant property that the HCF remains unchanged, ultimately leading to the desired result efficiently. The algorithm's efficiency stems from its iterative nature, quickly reducing the size of the numbers involved until the HCF is revealed.

Applications of HCF in Real-World Scenarios

Beyond the realm of abstract mathematics, the HCF finds practical applications in various fields:

• Simplifying Fractions: As mentioned earlier, HCF is crucial for simplifying fractions to their lowest terms. This improves clarity and facilitates calculations.
• Measurement and Cutting: Imagine you have two pieces of wood, one 120 cm long and the other 150 cm long. To cut them into pieces of equal length without any waste, you need to find the HCF of 120 and 150, which is 30 cm. You can cut each piece into 30 cm lengths.
• Array and Grid Arrangement: If you're arranging items in a rectangular grid, knowing the HCF helps determine the maximum size of identical squares that can be formed.
• Scheduling and Timing: In scheduling events or tasks that repeat at different intervals, the HCF helps determine the next time they will coincide.

Frequently Asked Questions (FAQ)

• What is the difference between HCF and LCM? The highest common factor (HCF) is the largest number that divides two or more numbers without leaving a remainder. The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. They are inversely related: HCF(a, b) x LCM(a, b) = a x b

• Can the HCF of two numbers be 1? Yes, if two numbers are coprime (they share no common factors other than 1), their HCF is 1.

• Which method is best for finding the HCF? For smaller numbers, listing factors or prime factorization may be quicker. For larger numbers, the Euclidean algorithm is significantly more efficient.

• What if I have more than two numbers? The methods described above can be extended to find the HCF of more than two numbers. For the prime factorization method, you find the prime factorization of each number and then identify the common prime factors with the lowest powers. For the Euclidean algorithm, you can find the HCF of two numbers, then find the HCF of the result and the next number, and so on.

Conclusion

Finding the HCF of 120 and 150, as demonstrated through prime factorization, listing factors, and the Euclidean algorithm, illustrates a fundamental concept in number theory. Understanding these methods provides a solid foundation for tackling more complex mathematical problems. The choice of method depends on the numbers involved and the desired level of efficiency. While the prime factorization method offers a clear visual understanding of the underlying principles, the Euclidean algorithm stands out as the most efficient method for larger numbers. Remember, the HCF is not just an abstract mathematical concept; it holds practical relevance in diverse real-world applications. Mastering the techniques to find the HCF enhances problem-solving skills and broadens mathematical comprehension.