Question 2.1. Determine all integer solutions of $172x + 20y = 1000$, such that $50 < x + y < 10$
Question 2.2. A farmer bought 100 livestock for a total cost of 4000 Euro. Calves cost 120
cach, Lambs 50 cach and Piglets 25 Euro cach. If the farmer obtained an even number of anim
each type, how many did he/she buy?
Question 2.3. Is 347 a prime integer? Justify your answer.
Question 2.4. Let $a, b, c, d, e, f, g, h, i, j \in \mathbb{N}$. Also let
$n = 2^2 \cdot 3^4 \cdot 11^a \cdot 13^b \cdot b^4$ and $m = 3^c \cdot 13^3 \cdot d^3 \cdot e^6$
such that
$\gcd(n, m) = 3^4 \cdot 13^f \cdot e^g$ and $\text{lcm}(n, m) = 3^5 \cdot 5^4 \cdot 11^2 \cdot 13^h \cdot 23^i \cdot e^j$
Determine a, b, c, d, e, f, g, h, i, j.
Question 2.5. Find the smallest $n \in \mathbb{N}$ such that $8^n \equiv 10 \pmod{11}$.
Question 2.6. Use modular arithmetic to answer the following:
(1) What is the remainder upon division of $3^{84}$ by 47?
(2) Does 17 divide $19^{18} - 16$?
