Let n be a positive integer and a complex number with unit modulus is a solution of the equation `Z^n+Z+1=0` , then the value of n can be

Options

A. 87
B. 97
C. 104
D. 222

Answer

To solve the equation \( Z^n + Z + 1 = 0 \) for a complex number \( Z \) with unit modulus, we will follow these steps:

### Step 1: Define the complex number
Let \( Z = \alpha \) be a complex number such that \( |\alpha| = 1 \). This means that \( \alpha \) can be expressed in the form:
\[ \alpha = e^{i\theta} \quad \text{for some angle } \theta. \]

### Step 2: Substitute into the equation
Substituting \( \alpha \) into the equation gives:
\[ \alpha^n + \alpha + 1 = 0. \]
Rearranging this, we find:
\[ \alpha^n = -\alpha - 1. \]

### Step 3: Take the modulus
Taking the modulus of both sides, we have:
\[ |\alpha^n| = |-\alpha - 1|. \]
Since \( |\alpha| = 1 \), we can simplify the left side:
\[ |\alpha^n| = |\alpha|^n = 1^n = 1. \]
Thus, we have:
\[ 1 = |-\alpha - 1|. \]

### Step 4: Simplify the right side
Now, we need to find \( |-\alpha - 1| \):
\[ |-\alpha - 1| = |-(\alpha + 1)| = |\alpha + 1|. \]
Thus, we have:
\[ 1 = |\alpha + 1|. \]

### Step 5: Analyze the modulus condition
The condition \( |\alpha + 1| = 1 \) implies that the point \( \alpha + 1 \) lies on the unit circle centered at the origin. This can be interpreted geometrically as follows:
- The point \( \alpha \) lies on the unit circle (since \( |\alpha| = 1 \)).
- The point \( -1 \) is located at \( (-1, 0) \) in the complex plane.

### Step 6: Geometric interpretation
The distance from \( -1 \) to \( \alpha \) must be 1. This means that the point \( \alpha \) must lie on the circle of radius 1 centered at \( -1 \). The center of this circle is at \( (-1, 0) \) and has a radius of 1.

### Step 7: Find conditions on \( n \)
The points \( \alpha \) that satisfy this condition are the solutions to the equation:
\[ |\alpha + 1| = 1. \]
This describes a circle in the complex plane. The angle \( \theta \) corresponding to \( \alpha \) must satisfy certain conditions based on \( n \).

### Step 8: Determine possible values of \( n \)
From the analysis, we find that \( n \) must be such that \( \alpha^n \) remains on the unit circle. This leads us to conclude that \( n \) must be a positive integer such that \( n \equiv 2 \mod 3 \).

### Conclusion
Thus, the possible values of \( n \) that satisfy the original equation are those that leave a remainder of 2 when divided by 3.