Understanding the Highest Common Factor (HCF)

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder. Finding the HCF is a fundamental concept in number theory and is useful in simplifying fractions and solving various mathematical problems.

To find the HCF of 96, 156, and 60, we can use the prime factorization method. This method involves breaking down each number into its prime factors. The HCF is then the product of the common prime factors, raised to the lowest power they appear in any of the factorizations.

Step-by-Step HCF Calculation for 96, 156, and 60

Let's find the prime factorization for each number: 96, 156, and 60.

Prime Factorization of 96:

• $96 = 2 \times 48$
• $48 = 2 \times 24$
• $24 = 2 \times 12$
• $12 = 2 \times 6$
• $6 = 2 \times 3$

So, the prime factorization of 96 is $2 \times 2 \times 2 \times 2 \times 2 \times 3$, which can be written as $2^5 \times 3^1$.

Prime Factorization of 156:

• $156 = 2 \times 78$
• $78 = 2 \times 39$
• $39 = 3 \times 13$

So, the prime factorization of 156 is $2 \times 2 \times 3 \times 13$, which can be written as $2^2 \times 3^1 \times 13^1$.

Prime Factorization of 60:

• $60 = 2 \times 30$
• $30 = 2 \times 15$
• $15 = 3 \times 5$

So, the prime factorization of 60 is $2 \times 2 \times 3 \times 5$, which can be written as $2^2 \times 3^1 \times 5^1$.

Identifying Common Prime Factors

Now, let's look for the prime factors that are common to all three numbers: 96, 156, and 60.

• The prime factor 2 is present in the factorization of 96 ($2^5$), 156 ($2^2$), and 60 ($2^2$). The lowest power of 2 among these is $2^2$.
• The prime factor 3 is present in the factorization of 96 ($3^1$), 156 ($3^1$), and 60 ($3^1$). The lowest power of 3 among these is $3^1$.
• The prime factor 5 is only present in 60.
• The prime factor 13 is only present in 156.

The common prime factors are 2 and 3. We take the lowest power of each common factor.

• Lowest power of 2: $2^2$
• Lowest power of 3: $3^1$

Calculating the HCF

To find the HCF, we multiply the common prime factors raised to their lowest powers:

$\text{HCF}(96, 156, 60) = 2^2 \times 3^1$

$\text{HCF}(96, 156, 60) = 4 \times 3$

$\text{HCF}(96, 156, 60) = 12$

Thus, the Highest Common Factor of 96, 156, and 60 is 12.

Revision Table: HCF Calculation Steps

Additional Information on HCF and Related Concepts

Besides prime factorization, the HCF of two numbers can also be found using the Euclidean Algorithm, which is particularly efficient for larger numbers. The Euclidean Algorithm is 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 one of the numbers becomes zero; the other number is the HCF.

Another related concept is the Least Common Multiple (LCM). The LCM of two or more numbers is the smallest positive integer that is a multiple of all the numbers. There is a relationship between HCF and LCM for two numbers, say 'a' and 'b': $\text{a} \times \text{b} = \text{HCF}(a, b) \times \text{LCM}(a, b)$. This relationship holds true only for two numbers, not generally for three or more.

Understanding HCF and LCM is crucial for operations with fractions, such as adding or subtracting fractions (finding a common denominator, which is related to LCM) and simplifying fractions (dividing numerator and denominator by their HCF).