Three sets of English, Mathematics and Science books containing 336, 240, 96 books respectively have to be stacked in such a way that all the books are stored subject - wise and the height of each stack is the same. Total number of stacks will be

Options

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

Answer

To solve the problem of how many total stacks can be made from the English, Mathematics, and Science books while keeping the height of each stack the same, we need to find the Highest Common Factor (HCF) of the number of books in each subject. Here’s a step-by-step solution:

### Step 1: Identify the number of books
- English books: 336
- Mathematics books: 240
- Science books: 96

### Step 2: Find the HCF of the three numbers
To find the HCF, we can use the prime factorization method.

1. **Prime Factorization of 336:**
- 336 = 2 × 168
- 168 = 2 × 84
- 84 = 2 × 42
- 42 = 2 × 21
- 21 = 3 × 7
- Therefore, 336 = 2^4 × 3^1 × 7^1

2. **Prime Factorization of 240:**
- 240 = 2 × 120
- 120 = 2 × 60
- 60 = 2 × 30
- 30 = 2 × 15
- 15 = 3 × 5
- Therefore, 240 = 2^4 × 3^1 × 5^1

3. **Prime Factorization of 96:**
- 96 = 2 × 48
- 48 = 2 × 24
- 24 = 2 × 12
- 12 = 2 × 6
- 6 = 2 × 3
- Therefore, 96 = 2^5 × 3^1

### Step 3: Determine the common factors
Now, we will take the lowest power of all common prime factors:
- For 2: The minimum power is 2^4 (from 336 and 240).
- For 3: The minimum power is 3^1 (common in all three).
- 5 and 7 are not common in all three.

### Step 4: Calculate the HCF
HCF = 2^4 × 3^1 = 16 × 3 = 48

### Step 5: Calculate the number of stacks for each subject
Now, we can find out how many stacks can be made for each subject by dividing the total number of books by the HCF.

1. **English books:**
- Number of stacks = 336 ÷ 48 = 7 stacks

2. **Mathematics books:**
- Number of stacks = 240 ÷ 48 = 5 stacks

3. **Science books:**
- Number of stacks = 96 ÷ 48 = 2 stacks

### Step 6: Calculate the total number of stacks
Total number of stacks = 7 (English) + 5 (Mathematics) + 2 (Science) = 14 stacks

### Final Answer
The total number of stacks will be **14**.

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