Question

The HCF of 96 and 432 is :

• A. 48
• B. 54
• C. 72
• D. 36

Hint:

For multiple-choice questions, you can also check the options starting from the largest one.
Test if the largest option divides both 96 and 432.
Here, $96 \div 72$ is not an integer, but $96 \div 48 = 2$ and $432 \div 48 = 9$. Thus, 48 is the highest common factor.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are required to find the Highest Common Factor (HCF) of the two numbers 96 and 432.

Step 2: Key Formula or Approach:
We will use the prime factorization method to find the HCF.
The steps for this method are:
1. Express each number as a product of its prime factors.
2. Identify the common prime factors.
3. The HCF is the product of the terms containing the lowest power of each common prime factor.
\[ \text{HCF}(a, b) = p_1^{\min(a_1, b_1)} \times p_2^{\min(a_2, b_2)} \times \dots \]

Step 3: Detailed Explanation:

• 1. Let us find the prime factorization of 96:
Dividing 96 continuously by prime numbers:
\[ 96 \div 2 = 48 \]
\[ 48 \div 2 = 24 \]
\[ 24 \div 2 = 12 \]
\[ 12 \div 2 = 6 \]
\[ 6 \div 2 = 3 \]
\[ 3 \div 3 = 1 \]
So, the prime factorization of 96 is:
\[ 96 = 2^5 \times 3^1 \]

• 2. Now, let us find the prime factorization of 432:
Dividing 432 continuously by prime numbers:
\[ 432 \div 2 = 216 \]
\[ 216 \div 2 = 108 \]
\[ 108 \div 2 = 54 \]
\[ 54 \div 2 = 27 \]
\[ 27 \div 3 = 9 \]
\[ 9 \div 3 = 3 \]
\[ 3 \div 3 = 1 \]
So, the prime factorization of 432 is:
\[ 432 = 2^4 \times 3^3 \]

• 3. Identify the common prime factors of 96 and 432, which are 2 and 3.

• 4. Determine the lowest power of each common prime factor:
- For prime factor 2: the powers are $2^5$ and $2^4$. The lower power is $2^4$.
- For prime factor 3: the powers are $3^1$ and $3^3$. The lower power is $3^1$.

• 5. Multiply these lowest powers to get the HCF:
\[ \text{HCF} = 2^4 \times 3^1 = 16 \times 3 = 48 \]


Step 4: Final Answer:
The Highest Common Factor (HCF) of 96 and 432 is 48, which corresponds to option (A).