A street shopkeeper prepares 396 Gulab jamuns and 342 ras-gullas. He packs them, in combination. Each container consists of either gulab jamuns or ras gullab but have equal number of pieces.
Find the numbers of pieces he should put in each box so that number of boxes are least.

Answer

To solve the problem of finding the number of pieces the shopkeeper should put in each box so that the number of boxes is minimized, we need to calculate the Highest Common Factor (HCF) of the two quantities: 396 gulab jamuns and 342 rasgullas.

### Step-by-Step Solution:

1. **Identify the Numbers**:
We have two numbers:
- Gulab Jamuns = 396
- Rasgullas = 342

2. **Use the Euclidean Algorithm to Find HCF**:
The Euclidean algorithm states that the HCF of two numbers can be found by repeatedly applying the formula:
\[ HCF(a, b) = HCF(b, a \mod b) \]
where \( a \) is the larger number and \( b \) is the smaller number.

3. **First Division**:
Divide 396 by 342:
\[ 396 \div 342 = 1 \quad \text{(quotient)} \]
Calculate the remainder:
\[ 396 - (342 \times 1) = 396 - 342 = 54 \]
So, we have:
\[ HCF(396, 342) = HCF(342, 54) \]

4. **Second Division**:
Now divide 342 by 54:
\[ 342 \div 54 = 6 \quad \text{(quotient)} \]
Calculate the remainder:
\[ 342 - (54 \times 6) = 342 - 324 = 18 \]
So, we have:
\[ HCF(342, 54) = HCF(54, 18) \]

5. **Third Division**:
Now divide 54 by 18:
\[ 54 \div 18 = 3 \quad \text{(quotient)} \]
Calculate the remainder:
\[ 54 - (18 \times 3) = 54 - 54 = 0 \]
Since the remainder is 0, we find that:
\[ HCF(54, 18) = 18 \]

6. **Conclusion**:
The HCF of 396 and 342 is 18. Therefore, the shopkeeper should put **18 pieces** in each box to minimize the number of boxes.

### Final Answer:
The number of pieces he should put in each box is **18**.

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