Find the HCF of `3^(25)+1,3^(35)+1` ?

Options

A. 3
B. 2
C. `3^(5)+1`
D. 7

Answer

To find the HCF of \(3^{25} + 1\) and \(3^{35} + 1\), we can use a mathematical property related to the HCF of numbers in the form \(a^m + 1\) and \(a^n + 1\).

### Step-by-Step Solution:

1. **Identify the numbers**: We have \(a = 3\), \(m = 25\), and \(n = 35\). We need to find the HCF of \(3^{25} + 1\) and \(3^{35} + 1\).

2. **Use the HCF formula**: The HCF of \(a^m + 1\) and \(a^n + 1\) can be calculated using the formula:
\[ \text{HCF}(a^m + 1, a^n + 1) = a^{\text{HCF}(m, n)} + 1 \]
Here, we need to find \(\text{HCF}(m, n)\), which is \(\text{HCF}(25, 35)\).

3. **Calculate HCF of 25 and 35**:
- The prime factorization of \(25\) is \(5^2\).
- The prime factorization of \(35\) is \(5 \times 7\).
- The common factor is \(5\).
- Thus, \(\text{HCF}(25, 35) = 5\).

4. **Substitute back into the formula**: Now we substitute back into the HCF formula:
\[ \text{HCF}(3^{25} + 1, 3^{35} + 1) = 3^{5} + 1 \]

5. **Calculate \(3^5 + 1\)**:
- First, calculate \(3^5 = 243\).
- Then, add \(1\): \(243 + 1 = 244\).

6. **Final answer**: Therefore, the HCF of \(3^{25} + 1\) and \(3^{35} + 1\) is \(244\).

### Summary:
The HCF of \(3^{25} + 1\) and \(3^{35} + 1\) is \(244\).

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