On Children's day celebrations , gifts were distributed among the children of the locality . There was a total of 96 pens and 72 pencils to be gives aways as gifts. What could be the total number of pens and pencils given to each chil if maximum number of children received the gifts and all gifts were identical ?

Options

A. 3
B. 4
C. 7
D. 12

Answer

To solve the problem step by step, we need to determine how many children can receive gifts such that each child gets the same number of pens and pencils. We will maximize the number of children receiving gifts by finding the highest common factor (HCF) of the total number of pens and pencils.

### Step 1: Identify the total number of gifts
We have:
- Total number of pens = 96
- Total number of pencils = 72

### Step 2: Find the HCF of 96 and 72
To maximize the number of children, we need to find the HCF of 96 and 72. We can do this by prime factorization.

**Prime Factorization of 72:**
1. 72 ÷ 2 = 36
2. 36 ÷ 2 = 18
3. 18 ÷ 2 = 9
4. 9 ÷ 3 = 3
5. 3 ÷ 3 = 1

So, the prime factorization of 72 is:
\[ 72 = 2^3 \times 3^2 \]

**Prime Factorization of 96:**
1. 96 ÷ 2 = 48
2. 48 ÷ 2 = 24
3. 24 ÷ 2 = 12
4. 12 ÷ 2 = 6
5. 6 ÷ 2 = 3
6. 3 ÷ 3 = 1

So, the prime factorization of 96 is:
\[ 96 = 2^5 \times 3^1 \]

### Step 3: Determine the common factors
Now, we can find the HCF by taking the lowest power of each common prime factor:
- For 2: The minimum power is \(2^3\)
- For 3: The minimum power is \(3^1\)

Thus, the HCF is:
\[ HCF = 2^3 \times 3^1 = 8 \times 3 = 24 \]

### Step 4: Calculate the number of gifts per child
Now that we know the maximum number of children (24), we can calculate how many pens and pencils each child will receive.

**Pencils per child:**
\[ \text{Pencils per child} = \frac{\text{Total Pencils}}{\text{Number of Children}} = \frac{72}{24} = 3 \]

**Pens per child:**
\[ \text{Pens per child} = \frac{\text{Total Pens}}{\text{Number of Children}} = \frac{96}{24} = 4 \]

### Step 5: Calculate the total number of gifts per child
Now, we can find the total number of gifts each child receives:
\[ \text{Total gifts per child} = \text{Pencils per child} + \text{Pens per child} = 3 + 4 = 7 \]

### Final Answer
The total number of gifts given to each child is **7**.

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