111 Russian, 407 Chinese and 259 Japanese tourists have assembled at a place Identical batches containing the same number of persons from each nationality has to be formed without leaving out anybody. Find the total number of persons in each batch.

Options

A. 18
B. 21
C. 10
D. 14

Answer

To solve the problem of forming identical batches of tourists from three different nationalities (Russian, Chinese, and Japanese), we need to follow these steps:

### Step 1: Identify the number of tourists from each nationality
- Russian tourists: 111
- Chinese tourists: 407
- Japanese tourists: 259

### Step 2: Calculate the total number of tourists
To find the total number of tourists, we add the number of tourists from each nationality:
\[ \text{Total tourists} = 111 + 407 + 259 \]
Calculating this gives:
\[ \text{Total tourists} = 777 \]

### Step 3: Find the highest common factor (HCF)
Next, we need to find the highest common factor (HCF) of the three numbers (111, 407, and 259). The HCF will tell us how many identical batches can be formed without leaving anyone out.

To find the HCF:
- The prime factorization of 111 is \(3 \times 37\).
- The prime factorization of 407 is \(11 \times 37\).
- The prime factorization of 259 is \(7 \times 37\).

The common factor among these is 37. Thus:
\[ \text{HCF}(111, 407, 259) = 37 \]

### Step 4: Calculate the number of tourists in each batch
Now, we can find the number of tourists in each batch. We do this by dividing the total number of tourists by the number of batches (which is the HCF we found):
\[ \text{Number of tourists in each batch} = \frac{\text{Total tourists}}{\text{Number of batches}} = \frac{777}{37} \]
Calculating this gives:
\[ \text{Number of tourists in each batch} = 21 \]

### Conclusion
Thus, the total number of persons in each batch is **21**.

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