Part a: Is 7 a Prime Gaussian Integer?

Gaussian Integers:

Gaussian integers are complex numbers of the form $a + bi$, where $a$ and $b$ are integers, and $i$ is the imaginary unit (where $i^2 = -1$). A Gaussian integer $z = a + bi$ is considered prime if it cannot be factored into the product of two non-unit Gaussian integers.

Prime Gaussian Integers:

A prime Gaussian integer is one of the following:

1. A Gaussian integer $z = a + bi$ where $a = 0$ or $b = 0$, and $|z|$ (the modulus of $z$) is a prime number congruent to 3 modulo 4.
2. A Gaussian integer $z = a + bi$ such that $|z|^2 = a^2 + b^2$ is a prime number in the usual sense.

Checking if $7$ is a Prime Gaussian Integer:

• $7$ can be written as $7 + 0i$, so $a = 7$ and $b = 0$.
• The modulus of $7 + 0i$ is $|7 + 0i| = \sqrt{7^2 + 0^2} = \sqrt{49} = 7$.

Now, check if $7$ is congruent to 3 modulo 4:

$7 \mod 4 = 3$

Since $7$ is a prime number congruent to 3 modulo 4, $7$ is a prime Gaussian integer.

Part b: Is $120 + 70i$ Divisible by $3 + i$?

To check if $120 + 70i$ is divisible by $3 + i$, we can perform the division in the Gaussian integers and check if the result is also a Gaussian integer.

Division in Gaussian Integers:

We divide two Gaussian integers $z_1$ and $z_2$ as follows:

$\frac{z_1}{z_2} = \frac{(120 + 70i)(3 - i)}{(3 + i)(3 - i)}$

Here, the conjugate $3 - i$ is used in the denominator to simplify the expression.

1. Denominator Calculation: $(3 + i)(3 - i) = 3^2 - i^2 = 9 - (-1) = 10$

2. Numerator Calculation: $(120 + 70i)(3 - i) = 120 \times 3 - 120 \times i + 70i \times 3 - 70i^2$ $= 360 - 120i + 210i - 70(-1)$ $= 360 + 90i + 70$ $= 430 + 90i$

So the division yields:

$\frac{120 + 70i}{3 + i} = \frac{430 + 90i}{10} = 43 + 9i$

Since $43 + 9i$ is a Gaussian integer, $120 + 70i$ is divisible by $3 + i$.

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Would you like more details or do you have any questions?

Here are some related questions:

1. What are Gaussian integers?
2. How do you determine the norm of a Gaussian integer?
3. Can all prime numbers be prime Gaussian integers?
4. What are the units in the Gaussian integers?
5. How do you multiply and divide Gaussian integers?
6. What is the significance of prime Gaussian integers?
7. Are there any other prime numbers that are prime Gaussian integers?
8. How can you verify if a complex number is a Gaussian integer?

Tip: When working with Gaussian integers, it's helpful to think about them in terms of both their algebraic properties (like divisibility) and their geometric interpretation in the complex plane.