78 books, 114 crayotis and 141 notebooks were distributed to school children equally. It was found that 6 were left undistributed in each. What was the total number of children?
A. 9
B. 8
C. 13
D. 12

Options

A. A
B. B
C. C
D. D

Answer

To solve the problem of distributing 78 books, 114 crayons, and 141 notebooks among school children with 6 items left undistributed in each category, we can follow these steps:

### Step 1: Calculate the number of items distributed
Since 6 items were left undistributed in each category, we need to subtract 6 from each total:

- Books distributed = 78 - 6 = 72
- Crayons distributed = 114 - 6 = 108
- Notebooks distributed = 141 - 6 = 135

### Step 2: Find the total number of items distributed
Now, we have the total number of items distributed:
- Total items distributed = Books distributed + Crayons distributed + Notebooks distributed
- Total items distributed = 72 + 108 + 135

Calculating this gives:
- Total items distributed = 315

### Step 3: Find the total number of children
To find the total number of children, we need to determine how many children can receive an equal number of items from the total distributed. Since we need to find the highest common factor (HCF) of the distributed items (72, 108, and 135), we will calculate the HCF.

#### Finding HCF:
1. **Prime factorization**:
- 72 = 2^3 × 3^2
- 108 = 2^2 × 3^3
- 135 = 3^3 × 5

2. **Identify common factors**:
- The common prime factor is 3.
- The lowest power of 3 in the factorizations is 3^2.

3. **Calculate HCF**:
- HCF = 3^2 = 9

### Conclusion
The total number of children is equal to the HCF we calculated, which is 9.

### Final Answer
The total number of children is **9**.

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