The problem describes a number converter which processes an input value $x$ and outputs a value $y$. The process involves taking the arithmetic square root of $x$, and the output $y$ depends on whether the square root is rational or irrational.

Let's analyze and answer each question step by step:

1. When the input value $x$ is 16, what is the output $y$?

   • Calculate the square root of 16: $\sqrt{16} = 4$
   • Since 4 is a rational number, the output $y$ is 4.

   $y = 4$

2. If the input values are valid but no $y$ value is output, what are all the possible values of $x$?

   • This implies that the square root of $x$ is irrational. Rational numbers include perfect squares (1, 4, 9, 16, etc.). Any $x$ that is not a perfect square will have an irrational square root. Thus, $x$ must be a non-perfect square positive integer (e.g., 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, etc.).

3. If the output $y$ is $\sqrt{5}$, write directly two values of $x$ that meet the requirement.

   • Since $y = \sqrt{5}$, the arithmetic square root of $x$ should yield $\sqrt{5}$. Thus, $x$ must be 5: $\sqrt{5} = \sqrt{5}$
   • Two such values of $x$ that meet this requirement are $5$ and any other value for which the square root would rationally yield $\sqrt{5}$. However, since the square root of 5 is unique and only $x = 5$ applies, $x = 5$ is the only valid value directly corresponding to $y = \sqrt{5}$.

Would you like any further details or have any questions?

Here are 5 questions you can ask:

1. How can we determine if a number has a rational or irrational square root?
2. What is the process of finding the square root of a non-perfect square?
3. How can you check if a number is a perfect square?
4. What are the properties of irrational numbers?
5. Can you explain more about the concept of arithmetic square roots?

Tip: When working with square roots, knowing whether the input is a perfect square can help you determine if the result will be a rational or irrational number.