Highest Common Factor Of 84 And 120

Finding the Highest Common Factor (HCF) of 84 and 120

This article will guide you through several methods to determine the highest common factor (HCF), also known as the greatest common divisor (GCD), of 84 and 120. Understanding HCF is crucial in various mathematical applications, from simplifying fractions to solving algebraic problems. We'll explore the prime factorization method and the Euclidean algorithm, providing a clear understanding of both approaches. By the end, you'll be able to confidently calculate the HCF of any two numbers.

What is the Highest Common Factor (HCF)?

The highest common factor (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 goes evenly into both numbers. For example, the HCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without any remainder.

Method 1: Prime Factorization

This method involves finding the prime factors of each number and then identifying the common factors. Let's apply this to 84 and 120:

1. Find the prime factors of 84: 84 = 2 x 42 = 2 x 2 x 21 = 2 x 2 x 3 x 7 = 2² x 3 x 7

2. Find the prime factors of 120: 120 = 2 x 60 = 2 x 2 x 30 = 2 x 2 x 2 x 15 = 2 x 2 x 2 x 3 x 5 = 2³ x 3 x 5

3. Identify common prime factors: Both 84 and 120 share the prime factors 2 and 3.

4. Calculate the HCF: Multiply the common prime factors raised to their lowest power. In this case, the lowest power of 2 is 2¹ and the lowest power of 3 is 3¹. Therefore, the HCF of 84 and 120 is 2 x 3 = 12.

Method 2: Euclidean Algorithm

The Euclidean algorithm is a more efficient method for larger numbers. It's based on the principle that the HCF of two numbers doesn't change if the larger number is replaced by its difference with the smaller number. This process is repeated until the two numbers are equal. Let's apply this to 84 and 120:

1. Start with the larger number (120) and the smaller number (84):

   120 = 1 x 84 + 36

2. Replace the larger number (120) with the remainder (36) and repeat:

   84 = 2 x 36 + 12

3. Repeat the process:

   36 = 3 x 12 + 0

4. The HCF is the last non-zero remainder: The last non-zero remainder is 12, so the HCF of 84 and 120 is 12.

Conclusion:

Both the prime factorization method and the Euclidean algorithm effectively determine the HCF of 84 and 120, resulting in the answer 12. The Euclidean algorithm is generally preferred for larger numbers due to its efficiency. Understanding these methods provides a strong foundation for tackling more complex mathematical problems involving factors and divisibility. Remember to practice these methods to improve your proficiency in calculating the highest common factor of any given numbers.