Congruences

Which of these statements are true? (Choose all that apply.)

What is the last digit of $3^{2026}$?

Find the inverse of $17$ modulo $43$: the $x \in \{1, \dots, 42\}$ with $17x \equiv 1 \pmod{43}$.

Find the smallest positive integer $x$ with $x \equiv 2 \pmod 5$, $x \equiv 3 \pmod 7$ and $x \equiv 4 \pmod 9$.

How many solutions modulo $36$ does the congruence $8x \equiv 12 \pmod{36}$ have?

The second proof of Fermat's theorem reaches $a^{p-1}(p-1)! \equiv (p-1)! \pmod p$ and then cancels $(p-1)!$. What justifies the cancellation?

Claim: "If $a^{n-1} \equiv 1 \pmod n$ for every $a$ coprime to $n$, then $n$ is prime." Which number is a counterexample?

Which Lean statement correctly formalises the existence half of the Chinese remainder theorem? (IsCoprime m n means $um + vn = 1$ for some $u, v$.)

Write crt(residues, moduli) that returns the smallest $x \ge 0$ with $x \equiv r_i \pmod{m_i}$ for all $i$, assuming the moduli are pairwise coprime. Combine the congruences one at a time, using an inverse modulo $m_i$ (Python's pow(a, -1, m) computes one). Then check Sunzi's answer.