1. **Problem statement:** We have 35 packages: some single ($x$) and the rest double ($35 - x$). Double packages cost 180 more than single packages. All double packages are sold, 4 single packages sold at full price $y$, and the rest single packages sold at a discount of 120 euros. Total revenue is 18480 euros. 2. **Setting up the problem:** Let $x$ be the total number of single packages. Let $y$ be the initial price of each single package. Number of double packages is $35 - x$. Price of double package is $y + 180$. 3. **Revenue equation:** Revenue from double packages: $(35 - x)(y + 180)$ Revenue from single packages: $4y + (x - 4)(y - 120)$ Total revenue: $$ (35 - x)(y + 180) + 4y + (x - 4)(y - 120) = 18480 $$ 4. **Simplify the revenue equation:** Expand terms: $$ (35 - x)y + (35 - x)180 + 4y + (x - 4)(y - 120) = 18480 $$ $$ 35y - xy + 6300 - 180x + 4y + xy - 4y - 120x + 480 = 18480 $$ Combine like terms: $$ 35y + 4y - 4y - xy + xy + 6300 + 480 - 180x - 120x = 18480 $$ $$ 35y + 6780 - 300x = 18480 $$ 5. **Rearranged equation:** $$ -300x + 35y = 18480 - 6780 $$ $$ -300x + 35y = 11700 $$ This is the Diophantine equation to solve. 6. **Solve the Diophantine equation:** Equation: $$ -300x + 35y = 11700 $$ Divide entire equation by 5: $$ -60x + 7y = 2340 $$ Rewrite: $$ 7y = 60x + 2340 $$ $$ y = \frac{60x + 2340}{7} $$ For $y$ to be integer, $60x + 2340$ must be divisible by 7. 7. **Find general solution:** Since $60 \equiv 4 \pmod{7}$ and $2340 \equiv 3 \pmod{7}$ (because $7 \times 334 = 2338$, remainder 2, so actually $2340 \equiv 2$), let's check carefully: $2340 \div 7 = 334$ remainder $2$, so $2340 \equiv 2 \pmod{7}$. So: $$ 4x + 2 \equiv 0 \pmod{7} $$ $$ 4x \equiv -2 \equiv 5 \pmod{7} $$ Multiply both sides by inverse of 4 mod 7, which is 2 (since $4 \times 2 = 8 \equiv 1 \pmod{7}$): $$ x \equiv 5 \times 2 = 10 \equiv 3 \pmod{7} $$ General solution for $x$: $$ x = 7k + 3, \quad k \in \mathbb{Z} $$ Corresponding $y$: $$ y = \frac{60(7k + 3) + 2340}{7} = \frac{420k + 180 + 2340}{7} = \frac{420k + 2520}{7} = 60k + 360 $$ 8. **Conditions for variables:** - $x$ must be positive and less than a third of 35 (since less than a third are single packages), so $x < \frac{35}{3} \approx 11.67$. - $y$ must be positive (price). - Number of single packages sold at discount is $x - 4$ and must be positive. 9. **Final answers:** - Diophantine equation: $$ -300x + 35y = 11700 $$ - General solution: $$ x = 7k + 3 $$ $$ y = 60k + 360 $$ - Conditions: $$ 0 < x < 12 $$ $$ y > 0 $$ $$ x - 4 > 0 $$ Hence, $k$ must be chosen so that these conditions hold.