Hcf Of 90 And 495

Finding the Highest Common Factor (HCF) of 90 and 495: 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 calculating the HCF of 90 and 495, exploring various methods and delving into the underlying mathematical principles. We'll move beyond a simple answer and explore the 'why' behind the calculations, making this a valuable resource for students and anyone looking to strengthen their understanding of number theory.

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. For example, the HCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 evenly. Understanding HCF is crucial for simplifying fractions, finding equivalent ratios, and solving various problems in algebra and number theory.

Method 1: Prime Factorization Method

This method involves breaking down each number into its prime factors and then identifying the common factors. Prime factorization is the process of expressing a number as a product of its prime factors (numbers divisible only by 1 and themselves).

Let's apply this to 90 and 495:

1. Prime Factorization of 90:

90 can be broken down as follows:

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

Therefore, the prime factorization of 90 is 2 x 3 x 3 x 5, or 2 x 3² x 5.

2. Prime Factorization of 495:

495 can be broken down as follows:

• 495 = 3 x 165
• 165 = 3 x 55
• 55 = 5 x 11

Therefore, the prime factorization of 495 is 3 x 3 x 5 x 11, or 3² x 5 x 11.

3. Identifying Common Factors:

Now, we compare the prime factorizations of 90 and 495:

90 = 2 x 3² x 5 495 = 3² x 5 x 11

The common prime factors are 3² and 5.

4. Calculating the HCF:

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

HCF(90, 495) = 3² x 5 = 9 x 5 = 45

Therefore, the HCF of 90 and 495 is 45. This means that 45 is the largest number that divides both 90 and 495 without leaving a remainder.

Method 2: Euclidean Algorithm

The Euclidean algorithm is an efficient method for finding the HCF of two numbers. 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 90 and 495:

1. Divide the larger number (495) by the smaller number (90):

495 ÷ 90 = 5 with a remainder of 45.

2. Replace the larger number with the remainder:

Now we find the HCF of 90 and 45.

3. Repeat the process:

90 ÷ 45 = 2 with a remainder of 0.

Since the remainder is 0, the HCF is the last non-zero remainder, which is 45.

Therefore, the HCF of 90 and 495, using the Euclidean algorithm, is 45. This method is particularly useful for larger numbers where prime factorization can become cumbersome.

Method 3: Listing Factors Method

This method involves listing all the factors of each number and then identifying the common factors. The largest common factor is the HCF.

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

2. Factors of 495: 1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495

3. Common Factors: 1, 3, 5, 9, 15, 45

4. Highest Common Factor: The largest common factor is 45.

Therefore, the HCF of 90 and 495, using the listing factors method, is 45. This method is straightforward but can be time-consuming for larger numbers with many factors.

Understanding the Significance of the HCF

The HCF (45) of 90 and 495 holds significant meaning. It represents the largest possible size of identical groups that can be formed from 90 and 495 items. For example, if you have 90 apples and 495 oranges, you could divide them into 45 groups, each containing 2 apples and 11 oranges.

Furthermore, the HCF is crucial for simplifying fractions. If you have a fraction like 90/495, you can simplify it by dividing both the numerator and denominator by their HCF (45):

90/495 = (90 ÷ 45) / (495 ÷ 45) = 2/11

This simplification makes the fraction easier to understand and work with.

Applications of HCF in Real-World Scenarios

The concept of HCF extends beyond abstract mathematical exercises. It finds practical applications in various real-world scenarios:

• Measurement and Cutting: Imagine you need to cut pieces of wood of lengths 90cm and 495cm into equal lengths without any waste. The HCF (45cm) will determine the longest possible length of the pieces.

• Resource Allocation: If you're distributing items among groups, the HCF helps ensure fair and equal distribution. For instance, if you have 90 pencils and 495 erasers to distribute equally amongst classes, the HCF (45) tells you that you can divide them among 45 classes.

• Fraction Simplification: As previously discussed, the HCF is fundamental for simplifying fractions, making calculations easier and facilitating a clearer understanding of ratios and proportions.

Frequently Asked Questions (FAQ)

• Q: What is the difference between HCF and LCM?

A: The Highest Common Factor (HCF) is the largest number that divides both numbers without leaving a remainder. The Least Common Multiple (LCM) is the smallest number that is a multiple of both numbers. They are inversely related; the product of the HCF and LCM of two numbers is always equal to the product of the two numbers.

• Q: Can the HCF of two numbers be one of the numbers?

A: Yes, if one number is a multiple of the other, the HCF will be the smaller number. For example, the HCF of 30 and 60 is 30.

• Q: What if the HCF of two numbers is 1?

A: If the HCF of two numbers is 1, they are considered relatively prime or coprime, meaning they share no common factors other than 1.

• Q: Which method is the best for finding the HCF?

A: The best method depends on the numbers involved. Prime factorization is easier for smaller numbers, while the Euclidean algorithm is more efficient for larger numbers. Listing factors is generally the least efficient for larger numbers.

Conclusion

Finding the HCF of 90 and 495, whether through prime factorization, the Euclidean algorithm, or listing factors, consistently yields the result of 45. This seemingly simple calculation reveals a deeper understanding of number theory and its practical applications. This article has not only provided the solution but also explored the underlying mathematical principles and real-world significance of finding the highest common factor. Mastering these concepts builds a strong foundation for more advanced mathematical studies and problem-solving. Remember to choose the method best suited to the numbers involved for efficiency and accuracy. The understanding of HCF is a valuable tool in your mathematical toolkit, applicable across various fields and scenarios.