1. **Problem:** Find the general integral solutions of the Diophantine equation $$5x + 8y = 1$$. 2. **Step 1:** Check if the equation has any integer solutions by verifying that $$ ext{gcd}(5,8)\mid 1$$. Since $$\text{gcd}(5,8) = 1$$ and 1 divides 1, solutions exist. 3. **Step 2:** Use the Extended Euclidean Algorithm to find one particular solution for $$5x + 8y = 1$$. - Divide 8 by 5: $$8 = 5 \times 1 + 3$$. - Divide 5 by 3: $$5 = 3 \times 1 + 2$$. - Divide 3 by 2: $$3 = 2 \times 1 + 1$$. - Since the remainder is 1, back substitute to express 1 as a combination of 5 and 8: $$1 = 3 - 2 \times 1$$ But $$2 = 5 - 3 \times 1$$, so: $$1 = 3 - (5 - 3 \times 1) = 2 \times 3 - 5$$ Also, $$3 = 8 - 5 \times 1$$, so: $$1 = 2 \times (8 - 5) - 5 = 2 \times 8 - 3 \times 5$$ 4. **Step 3:** The particular solution is $$x_0 = -3, y_0 = 2$$. 5. **Step 4:** The general solution for the equation $$5x + 8y = 1$$ is given by: $$x = x_0 + \frac{8}{1}t = -3 + 8t$$ $$y = y_0 - \frac{5}{1}t = 2 - 5t$$ where $$t$$ is any integer. 6. **Problem:** Find the general integral solutions of the Diophantine equation $$19x + 11y = 4$$. 7. **Step 1:** Check if solutions exist by verifying $$\text{gcd}(19,11)\mid 4$$. Since $$\text{gcd}(19,11) = 1$$ and 1 divides 4, solutions exist. 8. **Step 2:** Use Extended Euclidean Algorithm to find one particular solution for $$19x + 11y = 1$$: - $$19 = 11 \times 1 + 8$$ - $$11 = 8 \times 1 + 3$$ - $$8 = 3 \times 2 + 2$$ - $$3 = 2 \times 1 + 1$$ - Back substitute: $$1 = 3 - 2 \times 1$$ $$2 = 8 - 3 \times 2$$ Substitute back: $$1 = 3 - (8 - 3 \times 2) = 3 \times 3 - 8$$ $$3 = 11 - 8$$, so $$1 = (11 - 8) \times 3 - 8 = 3 \times 11 - 4 \times 8$$ $$8 = 19 - 11$$, so $$1 = 3 \times 11 - 4 \times (19 - 11) = 7 \times 11 - 4 \times 19$$ 9. **Step 3:** The particular solution for $$19x + 11y = 1$$ is $$x_0 = -4, y_0 = 7$$. 10. **Step 4:** Multiply the particular solution by 4 to find solution for $$19x + 11y = 4$$: $$x = -4 \times 4 = -16$$ $$y = 7 \times 4 = 28$$ 11. **Step 5:** The general solution for $$19x + 11y = 4$$ is: $$x = -16 + 11t$$ $$y = 28 - 19t$$ where $$t$$ is any integer. **Final answers:** - For $$5x + 8y = 1$$: $$x = -3 + 8t$$ $$y = 2 - 5t$$ - For $$19x + 11y = 4$$: $$x = -16 + 11t$$ $$y = 28 - 19t$$ where $$t\in \mathbb{Z}$$.