The number of distinct complex number(s) z, such that `|z|=1 and z^(3)` is purely imagninary, is/are equal to

Answer

To solve the problem, we need to find the number of distinct complex numbers \( z \) such that \( |z| = 1 \) and \( z^3 \) is purely imaginary.

### Step 1: Represent \( z \)
Let \( z \) be represented in polar form since \( |z| = 1 \). We can write:
\[ z = e^{i\theta} \]
where \( \theta \) is a real number.

### Step 2: Compute \( z^3 \)
Now, we compute \( z^3 \):
\[ z^3 = (e^{i\theta})^3 = e^{i3\theta} \]

### Step 3: Condition for \( z^3 \) to be purely imaginary
For \( z^3 \) to be purely imaginary, its real part must be zero. The real part of \( e^{i3\theta} \) is given by \( \cos(3\theta) \). Therefore, we need:
\[ \cos(3\theta) = 0 \]

### Step 4: Solve for \( \theta \)
The cosine function is zero at odd multiples of \( \frac{\pi}{2} \):
\[ 3\theta = \frac{\pi}{2} + n\pi \quad \text{for } n \in \mathbb{Z} \]
This gives:
\[ \theta = \frac{\pi}{6} + \frac{n\pi}{3} \]

### Step 5: Find distinct values of \( \theta \)
Now we need to find distinct values of \( \theta \) within one full rotation (from \( 0 \) to \( 2\pi \)):
- For \( n = 0 \): \( \theta = \frac{\pi}{6} \)
- For \( n = 1 \): \( \theta = \frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{6} + \frac{2\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2} \)
- For \( n = 2 \): \( \theta = \frac{\pi}{6} + \frac{2\pi}{3} = \frac{\pi}{6} + \frac{4\pi}{6} = \frac{5\pi}{6} \)
- For \( n = 3 \): \( \theta = \frac{\pi}{6} + \frac{3\pi}{3} = \frac{\pi}{6} + \pi = \frac{7\pi}{6} \)
- For \( n = 4 \): \( \theta = \frac{\pi}{6} + \frac{4\pi}{3} = \frac{\pi}{6} + \frac{8\pi}{6} = \frac{9\pi}{6} = \frac{3\pi}{2} \)
- For \( n = 5 \): \( \theta = \frac{\pi}{6} + \frac{5\pi}{3} = \frac{\pi}{6} + \frac{10\pi}{6} = \frac{11\pi}{6} \)

### Step 6: List distinct values of \( z \)
The distinct values of \( z \) corresponding to these angles are:
1. \( z_1 = e^{i\frac{\pi}{6}} \)
2. \( z_2 = e^{i\frac{\pi}{2}} \)
3. \( z_3 = e^{i\frac{5\pi}{6}} \)
4. \( z_4 = e^{i\frac{7\pi}{6}} \)
5. \( z_5 = e^{i\frac{3\pi}{2}} \)
6. \( z_6 = e^{i\frac{11\pi}{6}} \)

### Conclusion
Thus, the total number of distinct complex numbers \( z \) such that \( |z| = 1 \) and \( z^3 \) is purely imaginary is:
\[ \boxed{6} \]