1. **State the problem:** Solve the Diophantine equation $$45x + 30y = 15$$ for integers $x$ and $y$. 2. **Formula and rules:** A linear Diophantine equation in two variables $ax + by = c$ has integer solutions if and only if the greatest common divisor (gcd) of $a$ and $b$ divides $c$. 3. **Find gcd:** Calculate $$\gcd(45, 30)$$. $$45 = 3 \times 15$$ $$30 = 2 \times 3 \times 5$$ The gcd is $$15$$. 4. **Check divisibility:** Since $$15$$ divides $$15$$, solutions exist. 5. **Simplify equation:** Divide the entire equation by $$15$$: $$\frac{45}{15}x + \frac{30}{15}y = \frac{15}{15}$$ which simplifies to $$3x + 2y = 1$$ 6. **Find particular solution:** Use the Extended Euclidean Algorithm to solve $$3x + 2y = 1$$. We want integers $x_0, y_0$ such that $$3x_0 + 2y_0 = 1$$. Try $x=1$: $$3(1) + 2y = 1 \Rightarrow 3 + 2y = 1 \Rightarrow 2y = -2 \Rightarrow y = -1$$. So a particular solution is $$x_0 = 1, y_0 = -1$$. 7. **General solution:** The general solution to $$3x + 2y = 1$$ is $$x = x_0 + \frac{b}{d}t = 1 + \frac{2}{1}t = 1 + 2t$$ $$y = y_0 - \frac{a}{d}t = -1 - \frac{3}{1}t = -1 - 3t$$ where $t$ is any integer. 8. **Write final answer:** The solutions to the original equation $$45x + 30y = 15$$ are $$x = 1 + 2t$$ $$y = -1 - 3t$$ for any integer $t$. This means infinitely many integer solutions exist, parameterized by $t$.