Highest Common Factor Of 9 And 12

Finding the Highest Common Factor (HCF) of 9 and 12

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 guide you through several methods to determine the HCF of 9 and 12, explaining the process in detail so you understand the underlying principles. Understanding HCF is crucial for simplifying fractions, solving algebraic problems, and grasping more advanced mathematical concepts.

What is the Highest Common Factor (HCF)?

The HCF of two or more numbers is the largest number that divides evenly into all of the numbers without leaving a remainder. In simpler terms, it's the biggest number that is a factor of all the given numbers. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 9 are 1, 3, and 9.

Methods to Find the HCF of 9 and 12

There are several ways to calculate the HCF, including:

1. Listing Factors Method

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

• Factors of 9: 1, 3, 9
• Factors of 12: 1, 2, 3, 4, 6, 12

By comparing the lists, we can see that the common factors of 9 and 12 are 1 and 3. The largest of these common factors is 3. Therefore, the HCF of 9 and 12 is 3.

2. Prime Factorization Method

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

• Prime factorization of 9: 3 x 3 = 3²
• Prime factorization of 12: 2 x 2 x 3 = 2² x 3

The common prime factor is 3. The lowest power of 3 present in both factorizations is 3¹. Therefore, the HCF of 9 and 12 is 3.

3. Euclidean Algorithm

The Euclidean algorithm is a more efficient method for larger numbers. It's based on repeatedly applying the division algorithm until the remainder is 0. The last non-zero remainder is the HCF.

1. Divide the larger number (12) by the smaller number (9): 12 ÷ 9 = 1 with a remainder of 3.
2. Replace the larger number with the smaller number (9) and the smaller number with the remainder (3): 9 ÷ 3 = 3 with a remainder of 0.
3. Since the remainder is 0, the HCF is the last non-zero remainder, which is 3.

Conclusion:

Regardless of the method used, the highest common factor of 9 and 12 is 3. Understanding these different methods allows you to choose the most appropriate technique based on the numbers involved. The Euclidean algorithm, while slightly more complex, is particularly useful for finding the HCF of larger numbers. Mastering the concept of HCF is a key step in building a strong foundation in mathematics.