Statement 1: `2^(33)-1` is divisible by 7.
Statement 2: `x^(n)-a^(n)` is divisible by `x-a`, for all `n in NN` and `x ne a`.

Options

A. Statement 1 and Statement 2 are true and Statement 2 is the correct explanation for Statement 1
B. Statement 1 and Statement 2 are true but Statement 2 is not the correct explanation for Statement 1
C. Statement 1 is true but Statement 2 is false
D. Statement 2 is true but Statement 1 is false

Answer

To solve the problem, we need to analyze both statements and prove their validity step by step.

### Step 1: Analyze Statement 2
**Statement 2:** \( x^n - a^n \) is divisible by \( x - a \) for all \( n \in \mathbb{N} \) and \( x \neq a \).

**Proof by Mathematical Induction:**

1. **Base Case (n=1):**
- For \( n = 1 \):
\[ x^1 - a^1 = x - a \]
- Clearly, \( x - a \) is divisible by \( x - a \). Thus, the base case holds.

2. **Inductive Hypothesis:**
- Assume the statement is true for \( n = k \), i.e.,
\[ x^k - a^k \text{ is divisible by } x - a. \]

3. **Inductive Step (n=k+1):**
- We need to prove it for \( n = k + 1 \):
\[ x^{k+1} - a^{k+1} = x^{k} \cdot x - a^{k} \cdot a. \]
- We can rewrite this as:
\[ x^{k+1} - a^{k+1} = (x^k - a^k) \cdot x + a^k \cdot (x - a). \]
- By the inductive hypothesis, \( x^k - a^k \) is divisible by \( x - a \). Therefore, \( (x^k - a^k) \cdot x \) is also divisible by \( x - a \).
- The term \( a^k \cdot (x - a) \) is also clearly divisible by \( x - a \).
- Thus, the entire expression \( x^{k+1} - a^{k+1} \) is divisible by \( x - a \).

4. **Conclusion:**
- By the principle of mathematical induction, \( x^n - a^n \) is divisible by \( x - a \) for all \( n \in \mathbb{N} \).

### Step 2: Analyze Statement 1
**Statement 1:** \( 2^{33} - 1 \) is divisible by 7.

1. **Using Statement 2:**
- We can express \( 2^{33} - 1 \) as \( (2^3)^{11} - 1^{11} \).
- Here, let \( x = 2^3 = 8 \) and \( a = 1 \), and \( n = 11 \).
- According to Statement 2, since \( 8^{11} - 1^{11} \) is divisible by \( 8 - 1 = 7 \), we conclude that:
\[ 2^{33} - 1 \text{ is divisible by } 7. \]

### Final Conclusion:
Both statements are true:
- Statement 1 is true: \( 2^{33} - 1 \) is divisible by 7.
- Statement 2 is true: \( x^n - a^n \) is divisible by \( x - a \) for all \( n \in \mathbb{N} \) and \( x \neq a \).

### Answer:
Both Statement 1 and Statement 2 are true.

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