The greatest common divisor of `(3^3)^333 + 1 and (3^3)^334 + 1` is:
` 3^(3333)` +1 तथा `3^(3334)` +1 का महत्तम समापवर्तक कितना है?

Options

A. A. `3^(3333)` + 1
B. B. 20
C. C. 2
D. D. 1

Explanation

To find the greatest common divisor (GCD) of the two expressions \( (3^3)^{333} + 1 \) and \( (3^3)^{334} + 1 \), we can follow these steps:

### Step 1: Rewrite the expressions
We start by rewriting the expressions in simpler forms:
- Let \( a = 3^3 = 27 \).
- Then, we can rewrite the expressions as:
- \( a^{333} + 1 \)
- \( a^{334} + 1 \)

### Step 2: Identify the GCD
We need to find \( \text{GCD}(a^{333} + 1, a^{334} + 1) \).

### Step 3: Use the GCD property
We can use the property of GCD that states:
\[ \text{GCD}(x, y) = \text{GCD}(x, y - kx) \]
for any integer \( k \).

In our case, we can apply this property:
\[ \text{GCD}(a^{333} + 1, a^{334} + 1) = \text{GCD}(a^{333} + 1, (a^{334} + 1) - (a \cdot (a^{333} + 1))) \]

### Step 4: Simplify the second term
Calculating the second term:
\[ (a^{334} + 1) - a \cdot (a^{333} + 1) = a^{334} + 1 - a^{334} - a = 1 - a \]
Thus, we have:
\[ \text{GCD}(a^{333} + 1, 1 - a) \]

### Step 5: Substitute the value of \( a \)
Now substituting \( a = 27 \):
\[ 1 - a = 1 - 27 = -26 \]
So we need to find:
\[ \text{GCD}(a^{333} + 1, -26) \]

### Step 6: Calculate \( a^{333} + 1 \)
Now we calculate \( a^{333} + 1 \):
\[ a^{333} + 1 = 27^{333} + 1 \]
We need to check \( 27^{333} + 1 \) modulo 26.

### Step 7: Calculate \( 27 \mod 26 \)
Since \( 27 \equiv 1 \mod 26 \):
\[ 27^{333} \equiv 1^{333} \equiv 1 \mod 26 \]
Thus:
\[ 27^{333} + 1 \equiv 1 + 1 \equiv 2 \mod 26 \]

### Step 8: Find the GCD
Now we find:
\[ \text{GCD}(2, 26) = 2 \]

### Conclusion
Therefore, the greatest common divisor of \( (3^3)^{333} + 1 \) and \( (3^3)^{334} + 1 \) is:
\[ \boxed{2} \]