Hcf Of 90 And 252

Finding the Highest Common Factor (HCF) of 90 and 252: 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 comprehensively explore how to calculate the HCF of 90 and 252, using various methods, explaining the underlying principles, and providing practical examples to solidify your understanding. We will delve into both the traditional methods and more advanced techniques, ensuring you gain a complete grasp of this important mathematical concept.

Introduction: Understanding HCF and its Importance

The Highest Common Factor (HCF) of two or more numbers is the largest number that divides each of them without leaving a remainder. It's a crucial concept used in various mathematical applications, including simplifying fractions, solving algebraic equations, and understanding number theory. Understanding HCF is essential for anyone studying mathematics, from elementary school to advanced levels. In this article, we will focus on finding the HCF of 90 and 252, demonstrating several techniques to achieve this.

Method 1: Prime Factorization Method

This method involves finding the prime factorization of each number and then identifying the common prime factors raised to the lowest power.

Step 1: Prime Factorization of 90

90 can be expressed as a product of its prime factors:

90 = 2 × 45 = 2 × 3 × 15 = 2 × 3 × 3 × 5 = 2¹ × 3² × 5¹

Step 2: Prime Factorization of 252

Similarly, we find the prime factorization of 252:

252 = 2 × 126 = 2 × 2 × 63 = 2 × 2 × 3 × 21 = 2 × 2 × 3 × 3 × 7 = 2² × 3² × 7¹

Step 3: Identifying Common Prime Factors

Now, we compare the prime factorizations of 90 and 252:

90 = 2¹ × 3² × 5¹ 252 = 2² × 3² × 7¹

The common prime factors are 2 and 3.

Step 4: Calculating the HCF

The lowest power of the common prime factors is 2¹ and 3². Therefore, the HCF is:

HCF(90, 252) = 2¹ × 3² = 2 × 9 = 18

Therefore, the highest common factor of 90 and 252 is 18. This means 18 is the largest number that divides both 90 and 252 without leaving a remainder.

Method 2: Euclidean Algorithm

The Euclidean algorithm is a highly efficient method for finding the HCF of two 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, and that number is the HCF.

Step 1: Apply the Division Algorithm

We start by dividing the larger number (252) by the smaller number (90) and find the remainder:

252 = 90 × 2 + 72

Step 2: Replace the larger number with the remainder

Now, we replace the larger number (252) with the remainder (72) and repeat the process:

90 = 72 × 1 + 18

Step 3: Continue until the remainder is 0

We continue this process:

72 = 18 × 4 + 0

Since the remainder is now 0, the last non-zero remainder (18) is the HCF.

Therefore, the HCF(90, 252) = 18. The Euclidean algorithm provides a systematic and efficient way to compute the HCF, especially for larger numbers where prime factorization might become cumbersome.

Method 3: Listing Factors Method

This method involves listing all the factors of each number and then identifying the largest common factor. While straightforward for smaller numbers, it becomes less efficient for larger numbers.

Step 1: List Factors of 90

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

Step 2: List Factors of 252

Factors of 252: 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252

Step 3: Identify Common Factors

Comparing the lists, the common factors are: 1, 2, 3, 6, 9, 18.

Step 4: Determine the HCF

The largest common factor is 18. Therefore, the HCF(90, 252) = 18. While this method is conceptually simple, it's less efficient than prime factorization or the Euclidean algorithm for larger numbers.

Illustrative Examples and Applications

The HCF has practical applications in various situations:

• Simplifying Fractions: To simplify a fraction, we divide both the numerator and the denominator by their HCF. For example, to simplify the fraction 90/252, we divide both by their HCF (18), resulting in the simplified fraction 5/14.

• Dividing Quantities: Suppose you have 90 apples and 252 oranges, and you want to divide them into identical groups with the largest possible number of items in each group. The HCF (18) tells us you can create 18 identical groups, each containing 5 apples and 14 oranges.

• Geometry: Finding the greatest common divisor is crucial in geometry problems involving finding the largest possible square tile that can evenly cover a rectangular area. If the dimensions of the rectangle are 90 units and 252 units, the largest square tile would have side lengths equal to the HCF (18 units).

Frequently Asked Questions (FAQ)

• What if the HCF of two numbers is 1? If the HCF of two numbers is 1, they are called relatively prime or coprime. This means they share no common factors other than 1.

• Can the HCF of two numbers be larger than the smaller number? No, the HCF of two numbers can never be larger than the smaller of the two numbers.

• Which method is the most efficient? The Euclidean algorithm is generally the most efficient method, particularly for larger numbers, as it avoids the need for complete prime factorization.

• What if I have more than two numbers? The HCF can be extended to find the HCF of more than two numbers. You can use the Euclidean algorithm repeatedly, or find the prime factorization of all numbers and identify the common prime factors raised to the lowest power.

Conclusion

Finding the Highest Common Factor (HCF) is a fundamental skill in mathematics with broad applications. We have explored three different methods – prime factorization, the Euclidean algorithm, and listing factors – each offering a unique approach to solving this problem. While the listing factor method is suitable for smaller numbers, the prime factorization and especially the Euclidean algorithm are more efficient and adaptable for larger numbers. Understanding these methods provides a solid foundation for tackling more complex mathematical problems involving divisibility and number theory. Remember to choose the method that best suits your needs and the complexity of the numbers involved. The key takeaway is that understanding the concept of HCF and mastering these techniques will significantly enhance your mathematical skills and problem-solving capabilities.