Hcf Of 42 And 90

Finding the Highest Common Factor (HCF) of 42 and 90: 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 provides a comprehensive guide to finding the HCF of 42 and 90, exploring various methods and delving deeper into the underlying mathematical principles. Understanding HCF is crucial for simplifying fractions, solving algebraic equations, and understanding more advanced mathematical concepts. We will explore different techniques, from prime factorization to the Euclidean algorithm, providing a thorough understanding applicable to a wide range of numbers.

Introduction to Highest Common Factor (HCF)

The Highest Common Factor (HCF) of two or more numbers is the largest number that divides each of the numbers 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 leaving a remainder. Understanding HCF is essential in various mathematical applications, including simplifying fractions, solving equations, and working with ratios and proportions. This article focuses on finding the HCF of 42 and 90 using several methods.

Method 1: Prime Factorization

Prime factorization is a powerful method for finding the HCF of two or more numbers. It involves breaking down each number into its prime factors – numbers that are only divisible by 1 and themselves. Once we have the prime factors, we can identify the common factors and multiply them to find the HCF.

Let's apply this method to find the HCF of 42 and 90:

1. Prime Factorization of 42:

42 can be broken down as follows:

42 = 2 x 21 = 2 x 3 x 7

Therefore, the prime factors of 42 are 2, 3, and 7.

2. Prime Factorization of 90:

90 can be broken down as follows:

90 = 2 x 45 = 2 x 3 x 15 = 2 x 3 x 3 x 5 = 2 x 3² x 5

Therefore, the prime factors of 90 are 2, 3, 3, and 5.

3. Identifying Common Factors:

Comparing the prime factorizations of 42 (2 x 3 x 7) and 90 (2 x 3² x 5), we see that they share the prime factors 2 and 3.

4. Calculating the HCF:

To find the HCF, we multiply the common prime factors:

HCF(42, 90) = 2 x 3 = 6

Therefore, the highest common factor of 42 and 90 is 6.

Method 2: Listing Factors

This method involves listing all the factors of each number and then identifying the largest common factor. While simpler for smaller numbers, it can become cumbersome for larger numbers.

1. Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

2. Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

3. Common Factors: Comparing the two lists, the common factors are 1, 2, 3, and 6.

4. Highest Common Factor: The largest common factor is 6.

Therefore, the HCF of 42 and 90 is 6. 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 useful for larger numbers where prime factorization becomes more complex. 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, and that number is the HCF.

Let's apply the Euclidean algorithm to find the HCF of 42 and 90:

1. Step 1: Divide the larger number (90) by the smaller number (42) and find the remainder.

90 ÷ 42 = 2 with a remainder of 6.

2. Step 2: Replace the larger number (90) with the smaller number (42) and the smaller number with the remainder (6).

3. Step 3: Repeat the division process.

42 ÷ 6 = 7 with a remainder of 0.

4. Step 4: Since the remainder is 0, the HCF is the last non-zero remainder, which is 6.

Therefore, the HCF of 42 and 90 is 6. The Euclidean algorithm provides a systematic and efficient approach, particularly advantageous when dealing with larger numbers.

Mathematical Explanation and Applications of HCF

The HCF finds practical application in several areas:

• Simplifying Fractions: The HCF is used to simplify fractions to their lowest terms. For example, the fraction 42/90 can be simplified by dividing both the numerator and denominator by their HCF, which is 6, resulting in the simplified fraction 7/15.

• Solving Equations: HCF plays a vital role in solving Diophantine equations, which are equations where only integer solutions are sought.

• Measurement Problems: Determining the largest possible size of identical squares that can tile a rectangle uses the concept of HCF. If a rectangle has dimensions of 42 units and 90 units, the largest square that can tile it perfectly has a side length equal to the HCF of 42 and 90 (6 units).

• Modular Arithmetic: HCF is crucial in modular arithmetic and cryptography, particularly in finding modular inverses.

• Abstract Algebra: The concept extends to more abstract algebraic structures, where the equivalent of HCF (greatest common divisor) is defined for ideals in rings.

Frequently Asked Questions (FAQ)

Q1: What is the difference between HCF and LCM?

A1: The Highest Common Factor (HCF) is the largest number that divides both numbers without a remainder. The Least Common Multiple (LCM) is the smallest number that is a multiple of both numbers. They are related by the formula: HCF(a, b) x LCM(a, b) = a x b

Q2: Can the HCF of two numbers be 1?

A2: Yes, if two numbers have no common factors other than 1, their HCF is 1. Such numbers are called relatively prime or coprime.

Q3: Is there a limit to the size of numbers for which the HCF can be found?

A3: No, the methods described, particularly the Euclidean algorithm, can be used to find the HCF of arbitrarily large numbers, although computational limitations might exist for extremely large numbers.

Q4: Why is the Euclidean algorithm more efficient than prime factorization for large numbers?

A4: Prime factorization can become computationally expensive for large numbers. The Euclidean algorithm avoids the need for complete prime factorization, making it significantly more efficient.

Conclusion

Finding the Highest Common Factor (HCF) is a fundamental mathematical operation with diverse applications. We've explored three different methods – prime factorization, listing factors, and the Euclidean algorithm – demonstrating their effectiveness in determining the HCF of 42 and 90, which is 6. Understanding the HCF is crucial not just for solving simple mathematical problems but also for grasping more advanced concepts in various fields. The choice of method depends on the size of the numbers involved; while prime factorization and listing factors are suitable for smaller numbers, the Euclidean algorithm emerges as a more efficient and robust method for larger numbers. Mastering these methods empowers you with a valuable tool for various mathematical challenges.