The number of non-zero integral solution of the equation `| 1- 2i|^(x) \ 5^(x)` is

Options

A. Zero
B. One
C. Two
D. Three

Answer

To solve the equation \( |1 - 2i|^x = 5^x \) for the number of non-zero integral solutions, we will follow these steps:

### Step 1: Calculate the modulus of the complex number
We start by calculating the modulus of the complex number \( 1 - 2i \).

\[ |1 - 2i| = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

### Step 2: Rewrite the equation
Now we can rewrite the original equation using the modulus we found:

\[ (\sqrt{5})^x = 5^x \]

### Step 3: Simplify the equation
We can express \( \sqrt{5} \) as \( 5^{1/2} \). Thus, we can rewrite the left side:

\[ (5^{1/2})^x = 5^x \]

This simplifies to:

\[ 5^{x/2} = 5^x \]

### Step 4: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:

\[ \frac{x}{2} = x \]

### Step 5: Solve for \( x \)
To solve for \( x \), we can rearrange the equation:

\[ x - \frac{x}{2} = 0 \]

This simplifies to:

\[ \frac{x}{2} = 0 \implies x = 0 \]

### Step 6: Determine the number of non-zero integral solutions
Since we are looking for non-zero integral solutions, and we found that \( x = 0 \) is the only solution, we conclude that there are no non-zero integral solutions.

### Final Answer
Thus, the number of non-zero integral solutions of the equation is:

\[ \boxed{0} \]

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