`(2^(19) + 1)` is divisible by :

Options

A. 3
B. 4
C. 6
D. 3 & 6 both

Answer

To determine whether \(2^{19} + 1\) is divisible by a certain number, we can use the property of sums of powers.

### Step-by-Step Solution:

1. **Identify the expression**: We need to evaluate \(2^{19} + 1\).

2. **Recognize the form**: The expression can be recognized as a sum of two powers: \(a^n + b^n\) where \(a = 2\), \(b = 1\), and \(n = 19\).

3. **Use the divisibility rule**: There is a known result that states if \(n\) is an odd integer, then \(a^n + b^n\) is divisible by \(a + b\). Here, since \(n = 19\) is odd, we can apply this rule.

4. **Calculate \(a + b\)**: In our case, \(a + b = 2 + 1 = 3\).

5. **Conclusion**: Since \(2^{19} + 1\) is of the form \(a^n + b^n\) with \(n\) being odd, we conclude that \(2^{19} + 1\) is divisible by \(3\).

### Final Answer:
Thus, \(2^{19} + 1\) is divisible by \(3\).

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