Highest Common Factor Of 90 And 60

Finding the Highest Common Factor (HCF) of 90 and 60: A Deep Dive

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 fundamental in various mathematical applications, from simplifying fractions to solving algebraic problems. This article will explore multiple methods to determine the HCF of 90 and 60, providing a comprehensive understanding of the concept and its practical applications.

Method 1: Prime Factorization

The prime factorization method involves breaking down each number into its prime factors – numbers divisible only by 1 and themselves. Once we have the prime factorization of both numbers, we can identify the common prime factors and multiply them to find the HCF.

Step-by-step breakdown for 90:

1. Start with the smallest prime number, 2: 90 is an even number, so it's divisible by 2. 90 ÷ 2 = 45.
2. Continue with the next prime number, 3: 45 is divisible by 3. 45 ÷ 3 = 15.
3. 3 again: 15 is also divisible by 3. 15 ÷ 3 = 5.
4. Finally, 5: 5 is a prime number.

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

Step-by-step breakdown for 60:

1. Start with 2: 60 ÷ 2 = 30.
2. 2 again: 30 ÷ 2 = 15.
3. 3: 15 ÷ 3 = 5.
4. 5: 5 is a prime number.

Therefore, the prime factorization of 60 is 2 x 2 x 3 x 5, or 2² x 3 x 5.

Identifying the Common Factors:

Comparing the prime factorizations of 90 (2 x 3² x 5) and 60 (2² x 3 x 5), we can see that they share the following prime factors: 2, 3, and 5.

Calculating the HCF:

To find the HCF, we multiply the common prime factors: 2 x 3 x 5 = 30.

Therefore, the highest common factor of 90 and 60 is 30.

Method 2: Listing Factors

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.

Factors of 90:

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

Factors of 60:

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Common Factors:

Comparing the two lists, we identify the common factors: 1, 2, 3, 5, 6, 10, 15, 30.

Highest Common Factor:

The largest number in this list is 30. Therefore, the HCF of 90 and 60 is 30.

Method 3: Euclidean Algorithm

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

Step-by-step application to 90 and 60:

1. Start with the larger number (90) and the smaller number (60): 90 - 60 = 30. Now we have 60 and 30.

2. Repeat the process: 60 - 30 = 30. Now we have 30 and 30.

3. The numbers are equal: Since both numbers are now 30, the HCF is 30.

The Euclidean algorithm offers a systematic and efficient way to find the HCF, regardless of the size of the numbers involved.

Applications of HCF

The HCF has numerous practical applications across various fields:

• Simplifying Fractions: The HCF is used to simplify fractions to their lowest terms. For example, the fraction 90/60 can be simplified by dividing both the numerator and denominator by their HCF (30), resulting in the simplified fraction 3/2.

• Solving Word Problems: Many word problems in mathematics and real-world scenarios involve finding the HCF to determine the largest possible size or quantity. For example, if you have 90 apples and 60 oranges, and you want to distribute them into bags with the same number of apples and oranges in each bag, the HCF (30) tells you that you can make 30 bags, each containing 3 apples and 2 oranges.

• Geometry and Measurement: The HCF finds applications in geometry problems related to finding the greatest common measure of lengths or areas.

• Number Theory: The HCF is a crucial concept in number theory, forming the basis for further exploration of divisibility, prime numbers, and other number-theoretic properties.

Understanding the Concept of Divisibility

Before concluding, let's reinforce the fundamental concept of divisibility. A number 'a' is said to be divisible by another number 'b' if the division of 'a' by 'b' leaves no remainder (the remainder is 0). This means that 'b' is a factor of 'a'. The HCF identifies the largest factor common to a set of numbers.

Conclusion

Finding the highest common factor (HCF) of two or more numbers is a fundamental skill in mathematics with numerous applications. We've explored three effective methods: prime factorization, listing factors, and the Euclidean algorithm. While the listing factors method is straightforward for smaller numbers, the prime factorization and Euclidean algorithm methods are more efficient and scalable for larger numbers. Understanding the concept of HCF and its practical applications is crucial for success in various mathematical and real-world problems. Mastering these methods will equip you with a valuable tool for problem-solving and a deeper appreciation of number theory. Remember to practice these methods with various numbers to enhance your understanding and problem-solving skills. The more you practice, the faster and more accurately you'll be able to find the HCF of any given set of numbers.