Highest Common Factor Of 36 And 90

Finding the Highest Common Factor (HCF) of 36 and 90: A Comprehensive Guide

The highest common factor (HCF), also known as the greatest common divisor (GCD), is the largest number that divides exactly into two or more numbers without leaving a remainder. Understanding how to find the HCF is crucial in various mathematical applications, from simplifying fractions to solving algebraic problems. This article delves deep into finding the HCF of 36 and 90, exploring multiple methods and providing a comprehensive understanding of the concept.

Understanding the Concept of HCF

Before we jump into calculating the HCF of 36 and 90, let's solidify our understanding of the underlying concept. The HCF represents the largest positive integer that perfectly divides both 36 and 90. This means that when you divide 36 and 90 by the HCF, the result in both cases is a whole number (no remainder).

Think of it like finding the biggest tile you can use to completely cover a floor with dimensions 36 units and 90 units. The size of that tile represents the HCF.

Method 1: Prime Factorization Method

This is a classic and widely used method for finding the HCF. It involves breaking down each number into its prime factors – the smallest prime numbers that multiply to give the original number.

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, compare the prime factorizations of 36 and 90:

36 = 2² x 3² 90 = 2 x 3² x 5

The common factors are 2 (to the power of 1, as it's only present once in 90) and 3² (as both numbers have at least two 3s).

Step 4: Calculating the HCF

Multiply the common 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 Method

This method is more suitable for smaller numbers. It 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

Compare the two lists and identify the common factors: 1, 2, 3, 6, 9, 18

Step 4: Finding the Highest Common Factor

The largest number in the list of common factors is 18.

Therefore, the HCF(36, 90) = 18.

Method 3: Euclidean Algorithm

The Euclidean algorithm is a highly efficient method for finding the HCF of two numbers, especially for larger 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.

Step 1: Repeated Subtraction

Let's start with 90 and 36:

90 - 36 = 54

Now we have 36 and 54. Repeat:

54 - 36 = 18

Now we have 18 and 36. Repeat:

36 - 18 = 18

Now both numbers are 18.

Step 2: Determining the HCF

The HCF is the number that remains after the repeated subtraction: 18.

Therefore, the HCF(36, 90) = 18. This method can also be implemented using the modulo operator (%) which represents the remainder after division.

Method 4: Euclidean Algorithm using Modulo Operator

This is a more concise version of the Euclidean algorithm. Instead of repeated subtraction, we use the modulo operator (%) to find the remainder.

Step 1: Apply the Modulo Operator

90 % 36 = 18 (90 divided by 36 leaves a remainder of 18)

Now we replace the larger number (90) with the remainder (18):

36 % 18 = 0 (36 divided by 18 leaves a remainder of 0)

Step 2: Determine the HCF

When the remainder is 0, the HCF is the divisor in the last step. In this case, the HCF is 18.

Therefore, the HCF(36, 90) = 18. This method is computationally efficient and is preferred for larger numbers.

Applications of HCF

The HCF has numerous applications across various mathematical fields and real-world scenarios. Some key examples include:

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

• Solving Word Problems: Many word problems involving division or sharing require finding the HCF to determine the largest possible equal groups or portions.

• Geometry: The HCF is used in geometric problems related to finding the dimensions of the largest square tile that can cover a rectangular area.

• Number Theory: HCF plays a significant role in number theory, particularly in understanding the relationships between numbers and their divisors.

• Computer Science: The Euclidean algorithm, used to find the HCF, is a fundamental algorithm in computer science, used in cryptography and other computational tasks.

Conclusion

Finding the highest common factor of two numbers is a fundamental concept in mathematics. This article explored four different methods – prime factorization, listing factors, Euclidean algorithm (both subtraction and modulo operator versions) – for determining the HCF of 36 and 90. Each method offers a unique approach, allowing you to choose the most suitable one depending on the context and the size of the numbers involved. Understanding these methods provides a solid foundation for tackling more complex mathematical problems and appreciating the practical applications of HCF in various fields. The consistent answer, regardless of the method, is 18, reinforcing the accuracy and reliability of these techniques. Mastering the concept of HCF opens doors to a deeper understanding of number theory and its practical applications.