To solve this, we need to determine the value of $z$ such that the expression $\dfrac{z}{5 + 12i}$ is minimal and results in a prime number. Let’s go step-by-step:

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Step 1: Understanding the Expression

The expression is:

$\dfrac{z}{5 + 12i}$

We are tasked with finding $z$ such that this expression, likely interpreted as a complex number, is minimal (in some sense) and prime.

In the context of complex numbers, a Gaussian prime is a prime element in the set of Gaussian integers $\mathbb{Z}[i]$, which are numbers of the form $a + bi$ where $a, b \in \mathbb{Z}$. A Gaussian prime is a number that cannot be factored into smaller Gaussian integers (other than trivial factors like $1$ or $i$).

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Step 2: Dividing by a Complex Number

To perform the division $\dfrac{z}{5 + 12i}$, we typically multiply by the conjugate of $5 + 12i$. The conjugate of $5 + 12i$ is $5 - 12i$. Therefore:

$\dfrac{z}{5 + 12i} = z \cdot \dfrac{5 - 12i}{(5 + 12i)(5 - 12i)}$

Now, compute the denominator:

$(5 + 12i)(5 - 12i) = 5^2 + 12^2 = 25 + 144 = 169$

Thus:

$\dfrac{z}{5 + 12i} = \dfrac{z (5 - 12i)}{169}$

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Step 3: Gaussian Primes and Minimality

For this fraction to be prime and minimal, $z$ must be chosen such that $z (5 - 12i)$ is a Gaussian prime. In the context of Gaussian integers, we need $z$ to be a Gaussian integer itself (i.e., $z = a + bi$).

To achieve minimality, a reasonable choice for $z$ is to make $z (5 - 12i)$ a Gaussian prime that has the smallest possible norm.

Let’s test some small Gaussian integers for $z$ to find such a case.

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Step 4: Choosing $z$ and Verifying Primality

After trying small Gaussian integers, the following value works:

$z = 1 + i$

Let’s compute $(1 + i)(5 + 12i)$ and verify if it results in a Gaussian prime.

Multiplication:

$(1 + i)(5 + 12i) = 5 + 12i + 5i + 12i^2$

Recall that $i^2 = -1$:

$= 5 + 12i + 5i - 12$

Combine like terms:

$= (5 - 12) + (12i + 5i) = -7 + 17i$

Checking Primality:

The Gaussian integer $-7 + 17i$ is a Gaussian prime because its norm $N$ is:

$N = (-7)^2 + 17^2 = 49 + 289 = 338$

This norm cannot be factored into smaller norms of Gaussian integers, confirming that $-7 + 17i$ is a Gaussian prime.

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Conclusion:

The minimal value of $z$ for which $\dfrac{z}{5 + 12i}$ is prime is:

$\boxed{1 + i}$

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Do you want further details or have any questions?

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Related Questions:

1. What are Gaussian primes, and how are they defined?
2. How do you determine the conjugate of a complex number?
3. How can you find the norm of a Gaussian integer?
4. What are the criteria for a Gaussian integer to be prime?
5. How do you multiply two complex numbers?

Tip:

When dividing complex numbers, always multiply by the conjugate to simplify the expression and eliminate the imaginary part from the denominator.