Divisibility, Euclid's algorithm and unique factorisation

Use Euclid's algorithm to compute $\gcd(252, 198)$.

Which pairs $(x, y)$ satisfy $5x + 3y = 1$? (Choose all that apply.)

For which of these numbers $d$ is it true that for all integers $a, b$: if $d \mid ab$ then $d \mid a$ or $d \mid b$? (Choose all that apply.)

Compute $\operatorname{lcm}(84, 90)$.

Which of these fractions have a terminating decimal expansion? (Choose all that apply.)

In the proof of Euclid's lemma we assume $p \mid ab$ and $p \nmid a$, and then say $\gcd(p, a) = 1$. Why?

With a $6$-litre jug, a $15$-litre jug and a tap, what is the smallest positive amount of water you can measure exactly (by filling, emptying and pouring between the jugs)?

In the Hilbert numbers $1, 5, 9, 13, 17, 21, \dots$ (numbers of the form $4k + 1$), which statement is true?

Which Lean statement correctly formalises Euclid's lemma for natural numbers?

Write extended_gcd(a, b) for natural numbers a, b (not both zero), returning a triple (g, x, y) with g == gcd(a, b) and a * x + b * y == g. Use the recursion: if b == 0 return (a, 1, 0); otherwise get (g, x, y) for (b, a % b) and back-substitute using a % b == a - (a // b) * b.