Three sets of English, Mathematics and science contain 336, 240 and 96 books respectively. They have to be arranged in such a manner that they are arranged subjectwise and their height will be same. Find the number of heaps?

Options

A. 14
B. 21
C. 22
D. 48

Answer

To find the number of heaps that can be arranged subject-wise with the same height using the given number of books in English, Mathematics, and Science, we need to calculate the Highest Common Factor (HCF) of the three numbers: 336, 240, and 96.

### Step-by-Step Solution:

1. **Prime Factorization of Each Number:**
- **For 336:**
- Divide by 2: 336 ÷ 2 = 168
- Divide by 2: 168 ÷ 2 = 84
- Divide by 2: 84 ÷ 2 = 42
- Divide by 2: 42 ÷ 2 = 21
- Divide by 3: 21 ÷ 3 = 7
- 7 is a prime number.
- So, the prime factorization of 336 is:
\[ 336 = 2^4 \times 3^1 \times 7^1 \]

- **For 240:**
- Divide by 2: 240 ÷ 2 = 120
- Divide by 2: 120 ÷ 2 = 60
- Divide by 2: 60 ÷ 2 = 30
- Divide by 2: 30 ÷ 2 = 15
- Divide by 3: 15 ÷ 3 = 5
- 5 is a prime number.
- So, the prime factorization of 240 is:
\[ 240 = 2^4 \times 3^1 \times 5^1 \]

- **For 96:**
- Divide by 2: 96 ÷ 2 = 48
- Divide by 2: 48 ÷ 2 = 24
- Divide by 2: 24 ÷ 2 = 12
- Divide by 2: 12 ÷ 2 = 6
- Divide by 2: 6 ÷ 2 = 3
- 3 is a prime number.
- So, the prime factorization of 96 is:
\[ 96 = 2^5 \times 3^1 \]

2. **Finding the HCF:**
- Now, we will take the lowest power of all common prime factors from the factorizations:
- For \(2\): The lowest power is \(2^4\) (from 336 and 240).
- For \(3\): The lowest power is \(3^1\) (common in all three).
- \(7\) and \(5\) are not common in all three numbers.
- Therefore, the HCF is:
\[ HCF = 2^4 \times 3^1 = 16 \times 3 = 48 \]

3. **Conclusion:**
- The number of heaps that can be arranged subject-wise with the same height is **48**.