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 with applications extending far beyond simple arithmetic. This article will provide a thorough explanation of how to find the HCF of 36 and 90, exploring multiple methods and delving into the underlying mathematical principles. Understanding HCF is crucial for simplifying fractions, solving algebraic equations, and even understanding more advanced concepts in number theory. We'll cover various methods, from prime factorization to the Euclidean algorithm, ensuring a comprehensive understanding for learners of all levels.

Understanding Highest Common Factor (HCF)

Before we dive into calculating the HCF of 36 and 90, let's solidify our understanding of the concept. The HCF of two or more numbers is the largest number that divides exactly into each of them without leaving a remainder. In simpler terms, it's the biggest number that is a common factor of all the given 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. Therefore, the highest common factor (HCF) of 12 and 18 is 6.

Method 1: Prime Factorization

The prime factorization method involves breaking down each number into its prime factors – numbers divisible only by 1 and themselves. Let's apply this to find the HCF of 36 and 90.

Step 1: Prime Factorization of 36

36 can be broken down as follows:

• 36 = 2 x 18
• 18 = 2 x 9
• 9 = 3 x 3

Therefore, the prime factorization of 36 is 2² x 3².

Step 2: 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 3² x 5.

Step 3: Identifying Common Factors

Now, let's compare the prime factorizations of 36 and 90:

36 = 2² x 3² 90 = 2 x 3² x 5

The common prime factors are 2 and 3².

Step 4: Calculating the HCF

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

HCF (36, 90) = 2 x 3² = 2 x 9 = 18

Therefore, the highest common factor of 36 and 90 is 18.

Method 2: Listing Factors

This method is suitable for smaller numbers and involves listing all the factors of each number and then identifying the largest common factor.

Step 1: List Factors of 36

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Step 2: List Factors of 90

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

Step 3: Identify Common Factors

Comparing the two 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 of 36 and 90 is 18.

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 cumbersome. 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 become equal, and that number is the HCF.

Step 1: Repeated Subtraction

Let's start with 90 and 36:

90 - 36 = 54

Now we have 54 and 36:

54 - 36 = 18

Now we have 36 and 18:

36 - 18 = 18

Since both numbers are now 18, the HCF is 18.

Step 2: Division Method (More Efficient Version of Euclidean Algorithm)

The division method streamlines the subtraction process. We repeatedly divide the larger number by the smaller number and take the remainder. The process continues until the remainder is 0. The last non-zero remainder is the HCF.

• Divide 90 by 36: 90 = 36 x 2 + 18
• Divide 36 by the remainder 18: 36 = 18 x 2 + 0

Since the remainder is 0, the HCF is the last non-zero remainder, which is 18.

Why is the HCF Important?

Understanding and calculating the HCF has practical applications across various mathematical areas:

• Simplifying Fractions: The HCF helps in reducing fractions to their simplest form. For example, the fraction 36/90 can be simplified by dividing both the numerator and denominator by their HCF (18), resulting in the equivalent fraction 2/5.

• Solving Algebraic Equations: The HCF is crucial in simplifying algebraic expressions and finding solutions to equations involving variables and numbers.

• Number Theory: HCF is a cornerstone concept in number theory, influencing more advanced topics like modular arithmetic and cryptography.

• Real-world Applications: Concepts related to HCF are used in various practical scenarios like dividing items equally, scheduling tasks, and solving problems involving proportions.

Frequently Asked Questions (FAQ)

Q1: What if the HCF of two numbers is 1?

A1: If the HCF of two numbers is 1, they are called relatively prime or coprime. This means they have no common factors other than 1.

Q2: Can the HCF be larger than the smaller number?

A2: No. The HCF can never be larger than the smallest of the two numbers.

Q3: Which method is best for finding the HCF?

A3: The best method depends on the numbers involved. For smaller numbers, listing factors or prime factorization might be quicker. For larger numbers, the Euclidean algorithm is far more efficient.

Q4: Can this method be used for more than two numbers?

A4: Yes, the prime factorization and Euclidean algorithm can be extended to find the HCF of more than two numbers. For prime factorization, you find the prime factors of all the numbers and then identify the common factors with the lowest power. For the Euclidean algorithm, you would find the HCF of two numbers first, and then find the HCF of that result and the third number, and so on.

Conclusion

Finding the highest common factor is a fundamental skill in mathematics with numerous applications. We've explored three distinct methods – prime factorization, listing factors, and the Euclidean algorithm – providing a comprehensive understanding of how to calculate the HCF, particularly for the numbers 36 and 90. Mastering these techniques is essential for building a solid foundation in mathematics and preparing for more advanced concepts. Remember to choose the method best suited to the numbers involved to maximize efficiency and understanding. The HCF, seemingly a simple concept, unlocks a world of mathematical possibilities.