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**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:
   \[
   \begin{cases} 
   5x \equiv -1 \pmod{7} \\
   |x| \leq 10 
   \end{cases}
   \]

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:
   \[
   \begin{cases} 
   \text{LCM}(a; b) = 72 \\
   \text{GCD}(a; b) = 6 
   \end{cases}
   \]

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