Consider the following statements :
1. `(omega^(10)+1)^(7)+omega=0`
2. `(omega^(105)+1)^(10)=p^(10)` for some prime number p where `omega ne1` is a cube root of unity.
Which of the above statements is/are correct ?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. neither 1 nor 2

Explanation

To solve the given problem, we will analyze each statement one by one.

### Statement 1:
\[ (\omega^{10} + 1)^7 + \omega = 0 \]

**Step 1: Simplify \(\omega^{10}\)**
Since \(\omega\) is a cube root of unity, we know that:
\[ \omega^3 = 1 \implies \omega^6 = 1 \implies \omega^9 = \omega^0 = 1 \]
Thus, we can reduce \(\omega^{10}\):
\[ \omega^{10} = \omega^{9} \cdot \omega = 1 \cdot \omega = \omega \]

**Step 2: Substitute \(\omega^{10}\) into the equation**
Now substituting \(\omega^{10}\) into the expression:
\[ (\omega + 1)^7 + \omega = 0 \]

**Step 3: Simplify \(\omega + 1\)**
Using the property of cube roots of unity:
\[ 1 + \omega + \omega^2 = 0 \implies \omega + 1 = -\omega^2 \]

**Step 4: Substitute \(-\omega^2\)**
Now substituting this into the equation:
\[ (-\omega^2)^7 + \omega = 0 \]

**Step 5: Calculate \((- \omega^2)^7\)**
Calculating \((- \omega^2)^7\):
\[ (-\omega^2)^7 = -\omega^{14} = -\omega^{3 \cdot 4 + 2} = -\omega^2 \]

**Step 6: Substitute back into the equation**
Now substituting this back into the equation:
\[ -\omega^2 + \omega = 0 \]

**Step 7: Rearranging the equation**
This simplifies to:
\[ \omega - \omega^2 = 0 \]
Factoring out \(\omega\):
\[ \omega(1 - \omega) = 0 \]
Since \(\omega \neq 0\), we have \(1 - \omega = 0\) which implies \(\omega = 1\), which contradicts the condition that \(\omega \neq 1\). Therefore, **Statement 1 is false**.

### Statement 2:
\[ (\omega^{105} + 1)^{10} = p^{10} \text{ for some prime number } p \]

**Step 1: Simplify \(\omega^{105}\)**
Using the property of cube roots of unity:
\[ \omega^{105} = \omega^{3 \cdot 35} = 1 \]

**Step 2: Substitute into the equation**
Now substituting this into the expression:
\[ (1 + 1)^{10} = 2^{10} \]

**Step 3: Identify \(p\)**
We can express \(2^{10}\) as:
\[ 2^{10} = p^{10} \text{ where } p = 2 \]
Since \(2\) is a prime number, **Statement 2 is true**.

### Conclusion
- Statement 1 is false.
- Statement 2 is true.

Thus, the correct answer is **Option B: 2 only**.

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