Here is the translation of the exercise:

Exercise 01 (05 points):

1. Determine the natural number $a$ such that $202_a = a_{10}$.

2. Prove by congruence that for all $n \in \mathbb{N}$, $6^{2n+1} + 9^{3n}$ is a multiple of 7.

3. Solve in $\mathbb{Z}/5\mathbb{Z}$ the equation $x^2 - 4x + 3 = 0$.

4. Solve in $\mathbb{Z} \times \mathbb{Z}$ the equation $11x - 35y = 6$.

5. Solve in $\mathbb{Z}$ the system of equations:

   5x \equiv -1 \pmod{7} \\ |x| \leq 10 \end{cases}$$

6. Prove that the following fraction is irreducible for every natural number $n$: $\frac{7n + 5}{3n + 2}$

7. Determine the pairs of natural numbers $(a; b)$ such that $a < b$ and:

   \text{LCM}(a; b) = 72 \\ \text{GCD}(a; b) = 6 \end{cases}$$