A complex number z with (Im)(z)=4 and a positive integer n be such that `z/(z+n)=4i`, then the value of n, is

Options

A. 4
B. 16
C. 17
D. 32

Answer

To solve the problem, we need to find the value of the positive integer \( n \) given that the complex number \( z \) has an imaginary part of 4 and satisfies the equation:

\[ \frac{z}{z+n} = 4i \]

### Step-by-Step Solution:

1. **Define the complex number \( z \)**:
Since the imaginary part of \( z \) is given as 4, we can express \( z \) as:
\[ z = x + 4i \]
where \( x \) is the real part of \( z \).

2. **Substitute \( z \) into the equation**:
Now, substitute \( z \) into the equation:
\[ \frac{x + 4i}{(x + 4i) + n} = 4i \]

3. **Simplify the denominator**:
The denominator can be rewritten as:
\[ z + n = x + n + 4i \]
Thus, the equation becomes:
\[ \frac{x + 4i}{x + n + 4i} = 4i \]

4. **Cross multiply**:
Cross multiplying gives us:
\[ x + 4i = 4i(x + n + 4i) \]

5. **Expand the right-hand side**:
Expanding the right-hand side:
\[ x + 4i = 4ix + 4in - 16 \]
(since \( i^2 = -1 \))

6. **Rearranging the equation**:
Rearranging gives:
\[ x + 4i = 4ix + 4in - 16 \]

7. **Separate real and imaginary parts**:
From the equation, we can separate the real and imaginary parts:
- Real part: \( x = -16 + 4in \)
- Imaginary part: \( 4 = 4x \)

8. **Solve for \( x \)**:
From the imaginary part:
\[ 4 = 4x \implies x = 1 \]

9. **Substituting \( x \) back**:
Substitute \( x = 1 \) into the real part equation:
\[ 1 = -16 + 4in \]
Rearranging gives:
\[ 4in = 1 + 16 \implies 4in = 17 \implies n = \frac{17}{4i} \]
Since \( n \) must be a positive integer, we can find \( n \) by isolating it:
\[ n = 17 \]

### Conclusion:
The value of \( n \) is:
\[ \boxed{17} \]