The greatest common divisor of
`3^(3^333) + 1and 3^(3^334) + 1` is :

Options

A. 2
B. 1
C. `3^(3^333)+ 1`
D. 20

Answer

To find the greatest common divisor (GCD) of \(3^{3^{333}} + 1\) and \(3^{3^{334}} + 1\), we can use the property of GCD for numbers of the form \(a^m + 1\) and \(a^n + 1\).

### Step 1: Identify the two numbers
Let:
- \(a = 3^{3^{333}}\)
- \(b = 3^{3^{334}}\)

We need to find:
\[ \text{GCD}(a + 1, b + 1) \]

### Step 2: Rewrite \(b\)
Notice that:
\[ b = 3^{3^{334}} = 3^{3 \cdot 3^{333}} = (3^{3^{333}})^3 = a^3 \]

### Step 3: Use the GCD property
We can use the property:
\[ \text{GCD}(x + 1, y + 1) = \text{GCD}(x, y) \]
if \(x\) and \(y\) are both odd. In our case, both \(a\) and \(b\) are odd since \(3^{3^{333}}\) and \(3^{3^{334}}\) are both odd numbers.

Thus, we can write:
\[ \text{GCD}(a + 1, b + 1) = \text{GCD}(a, b) \]

### Step 4: Calculate \(\text{GCD}(a, b)\)
Since \(b = a^3\), we have:
\[ \text{GCD}(a, b) = \text{GCD}(a, a^3) = a \]

### Step 5: Substitute back to find the GCD
Now, substituting back:
\[ \text{GCD}(3^{3^{333}} + 1, 3^{3^{334}} + 1) = 3^{3^{333}} + 1 \]

### Step 6: Conclude the result
Thus, the greatest common divisor of \(3^{3^{333}} + 1\) and \(3^{3^{334}} + 1\) is:
\[ \text{GCD}(3^{3^{333}} + 1, 3^{3^{334}} + 1) = 3^{3^{333}} + 1 \]

### Final Answer
The greatest common divisor is \(3^{3^{333}} + 1\).

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