Question

The HCF of 960 and 432 is :

• A. 48
• B. 54
• C. 72
• D. 36

Hint:

Euclid's division algorithm is generally faster for larger numbers compared to prime factorization, as finding prime factors of large numbers can be time-consuming.
Always write down the steps of division clearly to avoid simple arithmetic mistakes.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks us to compute the Highest Common Factor (HCF) of two positive integers, 960 and 432.
The HCF is the largest positive integer that divides both numbers without leaving a remainder.

Step 2: Key Formula or Approach:
We can find the HCF using either:
Method 1: Euclid's Division Algorithm (repeated division process).
Method 2: Prime Factorization Method.

Step 3: Detailed Explanation:
Let us use Euclid's Division Lemma/Algorithm:
For any two positive integers $a$ and $b$ (where $a \gt b$), we can write $a = bq + r$, where $0 \le r \lt b$.
Here, $a = 960$ and $b = 432$.

• Step 1: Divide 960 by 432:
\[ 960 = 432 \times 2 + 96 \]
The remainder is $96$, which is not zero.

• Step 2: Now, take the divisor 432 as the new dividend and the remainder 96 as the new divisor:
\[ 432 = 96 \times 4 + 48 \]
The remainder is $48$, which is not zero.

• Step 3: Now, take the divisor 96 as the new dividend and the remainder 48 as the new divisor:
\[ 96 = 48 \times 2 + 0 \]
The remainder has now become zero.

• Since the remainder is zero, the divisor at this final stage is the HCF of the two numbers.
Therefore, $\text{HCF}(960, 432) = 48$.

Let us double-check using Prime Factorization:

• Prime factorization of 960:
\[ 960 = 2^6 \times 3 \times 5 \]

• Prime factorization of 432:
\[ 432 = 2^4 \times 3^3 \]

• To find the HCF, take the product of the lowest power of each common prime factor:
The common prime factors are 2 and 3.
- Lowest power of 2 is $2^4$.
- Lowest power of 3 is $3^1$.
\[ \text{HCF} = 2^4 \times 3^1 = 16 \times 3 = 48 \]

Both methods confirm the HCF is 48.

Step 4: Final Answer:
The HCF of 960 and 432 is 48.
Hence, option (A) is correct.