Find the number of roots of the equation `z^(15) = 1` satisfying `|arg z| lt pi//2`.

Explanation

To find the number of roots of the equation \( z^{15} = 1 \) that satisfy the condition \( |\arg z| < \frac{\pi}{2} \), we can follow these steps:

### Step 1: Identify the roots of unity
The equation \( z^{15} = 1 \) represents the 15th roots of unity. The general form for the nth roots of unity is given by:
\[ z_k = e^{i \frac{2k\pi}{n}} \quad \text{for } k = 0, 1, 2, \ldots, n-1 \]
In our case, \( n = 15 \), so the roots are:
\[ z_k = e^{i \frac{2k\pi}{15}} \quad \text{for } k = 0, 1, 2, \ldots, 14 \]

### Step 2: Calculate the arguments of the roots
The arguments of these roots are:
\[ \arg(z_k) = \frac{2k\pi}{15} \]
for \( k = 0, 1, 2, \ldots, 14 \).

### Step 3: Determine the range of arguments
We need to find the values of \( k \) such that:
\[ |\arg(z_k)| < \frac{\pi}{2} \]
This translates to:
\[ -\frac{\pi}{2} < \frac{2k\pi}{15} < \frac{\pi}{2} \]

### Step 4: Solve the inequalities
1. For the left inequality:
\[ -\frac{\pi}{2} < \frac{2k\pi}{15} \implies -\frac{15}{2} < 2k \implies -\frac{15}{4} < k \]
Since \( k \) must be a non-negative integer, the smallest value for \( k \) is \( 0 \).

2. For the right inequality:
\[ \frac{2k\pi}{15} < \frac{\pi}{2} \implies 2k < \frac{15}{2} \implies k < \frac{15}{4} = 3.75 \]
Thus, the possible integer values for \( k \) are \( 0, 1, 2, 3 \).

### Step 5: Count the valid roots
The valid integer values of \( k \) that satisfy the inequalities are \( 0, 1, 2, 3 \). This gives us a total of \( 4 \) roots.

### Conclusion
Therefore, the number of roots of the equation \( z^{15} = 1 \) satisfying \( |\arg z| < \frac{\pi}{2} \) is \( 4 \).

---