To find the prime factorization of the complex number \( 116 + 138i \), we'll follow these steps:

### Step 1: Determine if \( 116 + 138i \) can be factored into simpler Gaussian integers.

A Gaussian integer is a complex number of the form \( a + bi \) where both \( a \) and \( b \) are integers. Prime factorization in the Gaussian integers involves finding Gaussian primes, which are irreducible elements in the ring of Gaussian integers.

### Step 2: Calculate the norm of the number.
The norm of a Gaussian integer \( a + bi \) is given by:
\[
N(a + bi) = a^2 + b^2
\]
For \( 116 + 138i \), the norm is:
\[
N(116 + 138i) = 116^2 + 138^2 = 13456 + 19044 = 32500
\]

### Step 3: Factor the norm.
The norm \( N(116 + 138i) = 32500 \) can be factored into prime numbers in the usual way:
\[
32500 = 2^2 \times 5^4 \times 13
\]

### Step 4: Identify potential Gaussian prime factors.
Since the norm is the product of primes, and since each prime norm corresponds to a Gaussian prime, we consider the primes dividing the norm:
- \( 2 \) has norm 2.
- \( 5 \) has norm 5.
- \( 13 \) has norm 13.

Gaussian primes can also have non-integer norms if they are sums of squares corresponding to prime numbers.

### Step 5: Perform trial division by Gaussian primes.
We now attempt to divide \( 116 + 138i \) by Gaussian primes corresponding to \( 2 \), \( 5 \), and \( 13 \).

1. **Dividing by \( 1 + i \) (norm 2)**:
   \[
   \frac{116 + 138i}{1 + i} = \frac{(116 + 138i)(1 - i)}{2} = \frac{116 - 116i + 138i - 138}{2} = \frac{-22 + 22i}{2} = -11 + 11i
   \]
   Hence, \( 116 + 138i = (1 + i)(-11 + 11i) \).

2. **Further factorization**:
   - Next, consider \( -11 + 11i \) which has a norm of \( (-11)^2 + 11^2 = 242 \).
   - The norm \( 242 = 2 \times 11^2 \).

   - We can factor \( -11 + 11i \) by recognizing it as \( -11(1 - i) \).

Thus, the prime factorization in the Gaussian integers is:
\[
116 + 138i = (1 + i)(1 - i) \times (-11) \times 11 = (-11)(11)(2)(1 + i)(1 - i)
\]
The factors \( (1 + i) \) and \( (1 - i) \) are Gaussian primes with norms of 2.

### Final Prime Factorization:
The final prime factorization is:
\[
116 + 138i = 2 \times 11 \times (-11) \times (1 + i) \times (1 - i)
\]

This represents the product of Gaussian primes, including negative integers.

Would you like more details on any step or have any further questions?

### Related Questions:
1. What is the prime factorization of a complex number in general?
2. How can Gaussian primes be characterized?
3. What is the significance of the norm in Gaussian integers?
4. What are the rules for dividing complex numbers?
5. How does the prime factorization of complex numbers differ from integers?
6. Can you explain more about how to determine if a Gaussian integer is a prime?
7. How does the Euclidean algorithm apply in the context of Gaussian integers?
8. What is the importance of Gaussian integers in number theory?

### Tip:
When dealing with complex numbers, always consider the norms to guide the factorization process.

Explore the prime factorization of the complex number 116 + 138i using Gaussian integers. Understand the norms, Gaussian primes, and step-by-step factorization process. Suitable for advanced high school students and beyond.

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