Pandavas won a hen in the war of the Mahabharat. They brought it on the Ist January , 2002. This hen gave birth to the 7 new hens on the very first day. After it every new hen irrespective of its age everyday gave birth (only once in a lifetime) to 7 new hens. This process continued throughout the year, but no any hen had been died so far. On the 365th day all the Pandav shared equally all the hens among all the five brothers. The remaining (if these can not be shared equally) hens were donated to Krishna. The number of hens which the Krishna had received is :

Options

A. 3
B. 2
C. can't be determined
D. none of these

Answer

To solve the problem of how many hens the Pandavas had after one year and how many were left for Krishna, we can break it down step by step.

### Step 1: Understand the Initial Condition
On January 1, 2002, the Pandavas start with 1 hen.

### Step 2: Calculate the Number of Hens Each Day
- On the first day, the original hen gives birth to 7 new hens.
- Therefore, at the end of the first day, the total number of hens is:
\[ 1 + 7 = 8 \text{ hens} \]

### Step 3: Determine the Growth Pattern
- Each hen, including the new ones, will give birth to 7 new hens every day.
- This means that every hen alive at the beginning of the day will contribute to the total number of hens by the end of that day.

### Step 4: Create a Formula for Each Day
Let \( H(n) \) be the total number of hens at the end of day \( n \). The formula can be represented as:
\[ H(n) = H(n-1) + 7 \times H(n-1) \]
This simplifies to:
\[ H(n) = 8 \times H(n-1) \]
Starting with \( H(0) = 1 \) (the original hen), we can calculate the hens for each subsequent day.

### Step 5: Calculate the Number of Hens at the End of 365 Days
Using the formula, we can find:
\[ H(n) = 8^n \]
Thus, at the end of 365 days:
\[ H(365) = 8^{365} \]

### Step 6: Sharing Among the Pandavas
Now, we need to share the total number of hens among the 5 Pandavas. We calculate:
\[ \text{Hens per brother} = \frac{H(365)}{5} \]
And the remainder (hens left for Krishna) is:
\[ \text{Remaining hens} = H(365) \mod 5 \]

### Step 7: Calculate the Remainder
To find \( 8^{365} \mod 5 \), we can use properties of modular arithmetic:
\[ 8 \equiv 3 \mod 5 \]
Thus,
\[ 8^{365} \equiv 3^{365} \mod 5 \]
Using Fermat's Little Theorem, since \( 3^4 \equiv 1 \mod 5 \), we find:
\[ 365 \mod 4 = 1 \quad \Rightarrow \quad 3^{365} \equiv 3^1 \equiv 3 \mod 5 \]

### Final Step: Conclusion
The number of hens that Krishna received is:
\[ \text{Remaining hens} = 3 \]