Find HCF of `3^(3^(333))+1 and 3^(3^(334))+1`.

Answer

To find the HCF (Highest Common Factor) of the numbers \(3^{3^{333}} + 1\) and \(3^{3^{334}} + 1\), we can follow these steps:

### Step 1: Rewrite the second number
The second number can be rewritten using the property of exponents:
\[ 3^{3^{334}} = 3^{3^{333} \cdot 3} = (3^{3^{333}})^3 \]
Thus, we can express:
\[ 3^{3^{334}} + 1 = (3^{3^{333}})^3 + 1 \]

### Step 2: Use the sum of cubes formula
We can apply the sum of cubes formula, which states that:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
In our case, let \(a = 3^{3^{333}}\) and \(b = 1\). Therefore:
\[ (3^{3^{333}})^3 + 1^3 = (3^{3^{333}} + 1)((3^{3^{333}})^2 - 3^{3^{333}} \cdot 1 + 1^2) \]
This simplifies to:
\[ (3^{3^{333}} + 1)(3^{2 \cdot 3^{333}} - 3^{3^{333}} + 1) \]

### Step 3: Identify the common factor
Now we have two expressions:
1. \(3^{3^{333}} + 1\)
2. \((3^{3^{333}} + 1)(3^{2 \cdot 3^{333}} - 3^{3^{333}} + 1)\)

The common factor between these two expressions is clearly:
\[ 3^{3^{333}} + 1 \]

### Step 4: Conclusion
Thus, the HCF of \(3^{3^{333}} + 1\) and \(3^{3^{334}} + 1\) is:
\[ \text{HCF} = 3^{3^{333}} + 1 \]