Hcf Of 24 And 84

Finding the Highest Common Factor (HCF) of 24 and 84: 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. This article will delve into the various methods for determining the HCF of 24 and 84, explaining each step in detail and providing a deeper understanding of the underlying mathematical principles. We'll explore methods suitable for different skill levels, from elementary techniques to more advanced approaches. By the end, you'll not only know the HCF of 24 and 84 but also possess a robust understanding of how to find the HCF of any two numbers.

Understanding Highest Common Factor (HCF)

Before we begin calculating the HCF of 24 and 84, let's clarify what the term means. The HCF of two or more numbers is the largest number that divides exactly into each of them without leaving a remainder. It's the biggest common factor shared by those numbers. Think of it as the largest number that all the numbers are perfectly divisible by.

Method 1: Prime Factorization

This method involves breaking down each number into its prime factors—numbers that are only divisible by 1 and themselves. Then, we identify the common prime factors and multiply them together to find the HCF.

Step 1: Prime Factorization of 24

24 can be broken down as follows:

24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3 = 2³ × 3

Therefore, the prime factorization of 24 is 2³ × 3.

Step 2: Prime Factorization of 84

Now let's find the prime factors of 84:

84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7 = 2² × 3 × 7

The prime factorization of 84 is 2² × 3 × 7.

Step 3: Identifying Common Prime Factors

Comparing the prime factorizations of 24 (2³ × 3) and 84 (2² × 3 × 7), we see that they share the prime factors 2 and 3. Note that we take the lowest power of the common prime factors.

Step 4: Calculating the HCF

To find the HCF, we multiply the common prime factors raised to their lowest power:

HCF(24, 84) = 2² × 3 = 4 × 3 = 12

Therefore, the highest common factor of 24 and 84 is 12.

Method 2: Listing Factors

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

Step 1: Factors of 24

The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24.

Step 2: Factors of 84

The factors of 84 are: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.

Step 3: Identifying Common Factors

Comparing the lists, we find the common factors: 1, 2, 3, 4, 6, and 12.

Step 4: Determining the HCF

The largest common factor is 12. Therefore, the HCF of 24 and 84 is 12.

Method 3: Euclidean Algorithm

The Euclidean algorithm is a more efficient method for finding the HCF of 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. We repeatedly apply this principle until we reach a remainder of 0. The last non-zero remainder is the HCF.

Step 1: Divide the larger number (84) by the smaller number (24)

84 ÷ 24 = 3 with a remainder of 12.

Step 2: Replace the larger number with the remainder

Now we find the HCF of 24 and 12.

Step 3: Repeat the process

24 ÷ 12 = 2 with a remainder of 0.

Step 4: The HCF is the last non-zero remainder

Since the remainder is 0, the HCF is the last non-zero remainder, which is 12.

Why is the HCF Important?

Understanding and calculating the HCF has several practical applications in mathematics and beyond:

• Simplifying Fractions: The HCF helps in simplifying fractions to their lowest terms. For example, the fraction 24/84 can be simplified to 2/7 by dividing both the numerator and denominator by their HCF (12).

• Solving Word Problems: Many word problems involving division and common factors require finding the HCF to determine the optimal solution. For instance, determining the largest possible square tiles to cover a rectangular floor of dimensions 24 units by 84 units.

• Number Theory: The HCF is a fundamental concept in number theory, forming the basis for other advanced mathematical concepts.

• Cryptography: HCF plays a crucial role in certain cryptographic algorithms.

• Real-world Applications: Imagine you have 24 red marbles and 84 blue marbles. You want to create as many identical bags as possible, each with the same number of red and blue marbles. The HCF (12) tells you that you can create 12 bags, each containing 2 red marbles and 7 blue marbles.

Frequently Asked Questions (FAQ)

Q1: What if the HCF is 1?

If the HCF of two numbers is 1, it means they are relatively prime or coprime. This indicates that they share no common factors other than 1.

Q2: Can I use a calculator to find the HCF?

Many scientific calculators and online calculators have built-in functions to calculate the HCF of two or more numbers.

Q3: Which method is the best?

The best method depends on the numbers involved. For small numbers, listing factors is easiest. For larger numbers, the Euclidean algorithm is generally more efficient. Prime factorization is a powerful method that offers a deeper understanding of number properties.

Q4: What if I have more than two numbers?

To find the HCF of more than two numbers, you can apply any of the methods above iteratively. For example, to find the HCF of 24, 84, and 36, first find the HCF of 24 and 84 (which is 12), then find the HCF of 12 and 36 (which is 12). Therefore, the HCF of 24, 84, and 36 is 12.

Conclusion

Finding the HCF of 24 and 84, as demonstrated through various methods, provides a clear understanding of this fundamental mathematical concept. Whether you choose prime factorization, listing factors, or the Euclidean algorithm, the result remains the same: the HCF of 24 and 84 is 12. Understanding the HCF is crucial not only for solving mathematical problems but also for appreciating the underlying structure and relationships within numbers. Mastering this concept opens doors to more advanced mathematical explorations. Remember to choose the method that best suits your comfort level and the complexity of the numbers involved. Practice makes perfect, so keep practicing to improve your proficiency in finding the highest common factor of any given set of numbers.