Real numbers and irrationality

Which of these numbers are irrational? (Choose all that apply.)

Write $0.363636\ldots$ (the block $36$ repeating forever) as a fraction $\tfrac pq$ in lowest terms. What is the denominator $q$?

How many integers $x$ satisfy $|x - 3| \le 2$?

In the proof that $p^2 = 2q^2$ has no solution with $q \ne 0$, we write: "Since $p^2$ is even, $p$ is even." Which fact justifies this step?

A student copies the proof for $\sqrt 2$ to "prove" that $\sqrt 4$ is irrational: "Suppose $p^2 = 4q^2$ with $q \ge 1$ as small as possible. Then $p^2$ is divisible by $4$, so $p$ is divisible by $4$; write $p = 4k$. Then $16k^2 = 4q^2$, so $q^2 = 4k^2$ — a smaller solution." Since $\sqrt 4 = 2$, something is wrong. Which step fails?

Let $S = \{1 - \tfrac1n : n = 1, 2, 3, \dots\}$. Which statement is true?

Which pair $(x, y)$ is a counterexample to the claim "if $x$ and $y$ are irrational, then $x + y$ is irrational"?

Which Lean statement says that $\Q$ is not complete, using the set $S = \{q \in \Q : q^2 < 2\}$? (IsLUB S b means "$b$ is a least upper bound of $S$".)

Write repetend_length(p, q) for integers $0 \le p$ and $q \ge 1$: the length of the repeating block of the decimal expansion of $\tfrac pq$, or 0 if the decimal terminates. Use long division and remember the position at which each remainder first appeared. For example $\tfrac17 = 0.\overline{142857}$ gives 6, $\tfrac{5}{12} = 0.41\overline{6}$ gives 1, and $\tfrac38 = 0.375$ gives 0.