1. **State the problem:** Solve the linear Diophantine equation $$172x + 20y = 1000$$ for integers $x$ and $y$. 2. **Recall the condition for solutions:** A linear Diophantine equation $ax + by = c$ has integer solutions if and only if the greatest common divisor (gcd) of $a$ and $b$ divides $c$. 3. **Calculate gcd:** $$\gcd(172, 20) = 4$$ 4. **Check divisibility:** Since $4$ divides $1000$ (because $1000 \div 4 = 250$), solutions exist. 5. **Simplify the equation by dividing all terms by gcd:** $$172x + 20y = 1000 \implies \cancel{4}(43x) + \cancel{4}(5y) = \cancel{4}(250)$$ $$43x + 5y = 250$$ 6. **Use the Extended Euclidean Algorithm to find one particular solution to $43x + 5y = 1$:** - $43 = 5 \times 8 + 3$ - $5 = 3 \times 1 + 2$ - $3 = 2 \times 1 + 1$ - $2 = 1 \times 2 + 0$ Back-substitute to express 1 as a combination of 43 and 5: - $1 = 3 - 2 \times 1$ - $2 = 5 - 3 \times 1$ So, $$1 = 3 - (5 - 3 \times 1) = 2 \times 3 - 5$$ But $3 = 43 - 5 \times 8$, so $$1 = 2(43 - 5 \times 8) - 5 = 2 \times 43 - 16 \times 5 - 5 = 2 \times 43 - 17 \times 5$$ Thus, a particular solution to $43x + 5y = 1$ is: $$x_0 = 2, \quad y_0 = -17$$ 7. **Multiply this solution by 250 to solve $43x + 5y = 250$:** $$x = 2 \times 250 = 500, \quad y = -17 \times 250 = -4250$$ 8. **General solution:** Since the equation is linear, the general solution is: $$x = 500 + 5t, \quad y = -4250 - 43t$$ where $t$ is any integer. **Final answer:** $$\boxed{x = 500 + 5t, \quad y = -4250 - 43t, \quad t \in \mathbb{Z}}$$