Hcf Of 36 And 90

Finding the Highest Common Factor (HCF) of 36 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 will delve into the various methods of calculating the HCF of 36 and 90, explaining each step in detail and providing a deeper understanding of the underlying principles. We'll cover prime factorization, the Euclidean algorithm, and explore the significance of HCF in different mathematical applications. Understanding HCF is crucial for simplifying fractions, solving algebraic equations, and tackling more advanced mathematical problems. This guide will equip you with the knowledge and skills to confidently determine the HCF of any two numbers.

Understanding 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 is a factor of both numbers. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The common factors of 12 and 18 are 1, 2, 3, and 6. The highest of these common factors is 6, therefore, the HCF of 12 and 18 is 6.

Now, let's focus on finding the HCF of 36 and 90.

Method 1: Prime Factorization

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

Step 1: Prime Factorization of 36

36 can be broken down into its prime factors as follows:

36 = 2 x 18 = 2 x 2 x 9 = 2 x 2 x 3 x 3 = 2² x 3²

Step 2: Prime Factorization of 90

90 can be broken down into its prime factors as follows:

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

Step 3: Identifying Common Prime Factors

Both 36 and 90 share the prime factors 2 and 3.

Step 4: Calculating the HCF

The lowest power of the common prime factor 2 is 2¹, and the lowest power of the common prime factor 3 is 3¹. Therefore, the HCF of 36 and 90 is 2 x 3 = 18.

Method 2: Listing Factors

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

Step 1: Listing Factors of 36

The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36

Step 2: Listing Factors of 90

The factors of 90 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Step 3: Identifying Common Factors

The common factors of 36 and 90 are: 1, 2, 3, 6, 9, 18

Step 4: Determining the HCF

The largest common factor is 18. Therefore, the HCF of 36 and 90 is 18.

Method 3: The 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 does not change if the larger number is replaced by its difference with the smaller number. This process is repeated until the two numbers become equal, and that number is the HCF.

Step 1: Divide the larger number (90) by the smaller number (36)

90 ÷ 36 = 2 with a remainder of 18

Step 2: Replace the larger number with the remainder

Now we find the HCF of 36 and 18.

Step 3: Repeat the division

36 ÷ 18 = 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 18. Therefore, the HCF of 36 and 90 is 18.

Explanation of the Euclidean Algorithm's Efficiency

The Euclidean algorithm is remarkably efficient because it avoids the need to find all the factors of the numbers involved. It systematically reduces the problem to smaller and smaller numbers until the HCF is revealed. This makes it particularly useful when dealing with very large numbers where the prime factorization method might become computationally expensive.

Applications of HCF

The concept of HCF has numerous applications across various fields of mathematics and beyond:

• Simplifying Fractions: The HCF is used to simplify fractions to their lowest terms. For instance, the fraction 36/90 can be simplified by dividing both the numerator and the denominator by their HCF, which is 18. This simplifies the fraction to 2/5.

• Algebraic Expressions: Finding the HCF of coefficients in algebraic expressions is crucial for factoring and simplifying these expressions.

• Measurement Problems: The HCF is used to determine the largest possible size of identical squares or rectangles that can tile a given area. For example, if you have a rectangular piece of land measuring 36 meters by 90 meters, you can divide it into identical squares of 18 meters by 18 meters.

• Number Theory: The HCF is fundamental in various areas of number theory, including modular arithmetic and cryptography.

• Computer Science: Algorithms related to HCF are used in computer programming for various applications like data compression and encryption.

Frequently Asked Questions (FAQ)

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

A: If the HCF of two numbers is 1, the numbers are said to be relatively prime or coprime. This means they have no common factors other than 1.

Q: Can the HCF of two numbers be larger than the smaller number?

A: No. The HCF of two numbers is always less than or equal to the smaller of the two numbers.

Q: Is there a difference between HCF and GCD?

A: No, HCF (Highest Common Factor) and GCD (Greatest Common Divisor) are two different names for the same concept.

Conclusion

Finding the Highest Common Factor (HCF) of two numbers is a valuable skill with broad applications in mathematics and other fields. We've explored three different methods – prime factorization, listing factors, and the Euclidean algorithm – each offering a unique approach to solving this problem. The Euclidean algorithm, in particular, stands out for its efficiency, especially when dealing with larger numbers. Understanding these methods and the underlying principles allows for a deeper comprehension of mathematical concepts and their practical applications. Remember to choose the method that best suits your needs and comfort level, but always strive for understanding the underlying mathematical reasoning. Mastering HCF will pave the way for tackling more complex mathematical challenges with confidence.