Three sets of English, mathematics and science books containing , 336, 240 and 96 books respectively have to be stacked in such a way that all the books are stored subjectwise and the height of each stack is the same. How many stacks will be there ?

Explanation

To solve the problem of stacking the books in such a way that all stacks are of equal height, 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 in each subject
- 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.

#### Prime Factorization:
1. **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: \( 2^4 \times 3^1 \times 7^1 \)

2. **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: \( 2^4 \times 3^1 \times 5^1 \)

3. **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: \( 2^5 \times 3^1 \)

### Step 3: Determine the common factors
Now, we take the lowest power of all common prime factors:
- For 2: The minimum power is \( 2^4 \)
- For 3: The minimum power is \( 3^1 \)

Thus, the HCF is:
\[ HCF = 2^4 \times 3^1 = 16 \times 3 = 48 \]

### Step 4: Calculate the number of stacks
Now that we know the height of each stack is 48 books, we can find the number of stacks for each subject:
1. For English: \( \frac{336}{48} = 7 \)
2. For Mathematics: \( \frac{240}{48} = 5 \)
3. For Science: \( \frac{96}{48} = 2 \)

### Step 5: Total number of stacks
Now, we add the number of stacks for each subject:
\[ \text{Total stacks} = 7 + 5 + 2 = 14 \]

### Final Answer:
There will be a total of **14 stacks**.

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