1. **Problem:** Solve the linear Diophantine equation $$172x + 20y = 1000$$ completely. 2. **Step 1: Find the gcd of 172 and 20.** We use the Euclidean algorithm: $$172 = 20 \times 8 + 12$$ $$20 = 12 \times 1 + 8$$ $$12 = 8 \times 1 + 4$$ $$8 = 4 \times 2 + 0$$ So, $$\gcd(172, 20) = 4$$. 3. **Step 2: Check if the equation has solutions.** A linear Diophantine equation $$ax + by = c$$ has solutions if and only if $$\gcd(a,b) \mid c$$. Here, $$4 \mid 1000$$ since $$1000/4 = 250$$, so solutions exist. 4. **Step 3: Simplify the equation by dividing all terms by gcd 4:** $$\frac{172}{4}x + \frac{20}{4}y = \frac{1000}{4}$$ $$43x + 5y = 250$$ 5. **Step 4: Solve the simplified equation using the Extended Euclidean Algorithm to find particular solution for $$x$$ and $$y$$.** We want integers $$x_0, y_0$$ such that: $$43x_0 + 5y_0 = 1$$ Then multiply by 250 to get solution for original equation. Using Extended Euclidean Algorithm: $$43 = 5 \times 8 + 3$$ $$5 = 3 \times 1 + 2$$ $$3 = 2 \times 1 + 1$$ $$2 = 1 \times 2 + 0$$ Back-substitute to express 1 as combination of 43 and 5: $$1 = 3 - 2 \times 1$$ $$= 3 - (5 - 3 \times 1) = 2 \times 3 - 5$$ $$= 2 \times (43 - 5 \times 8) - 5 = 2 \times 43 - 17 \times 5$$ So, $$x_0 = 2, y_0 = -17$$ satisfy $$43x_0 + 5y_0 = 1$$. 6. **Step 5: Multiply by 250 to get particular solution:** $$x_p = 2 \times 250 = 500$$ $$y_p = -17 \times 250 = -4250$$ 7. **Step 6: Write the general solution:** Since $$\gcd = 4$$, general solutions are: $$x = x_p + \frac{5}{4}t = 500 + 5t$$ $$y = y_p - \frac{43}{4}t = -4250 - 43t$$ where $$t$$ is any integer. **Final answer:** $$\boxed{x = 500 + 5t, \quad y = -4250 - 43t, \quad t \in \mathbb{Z}}$$