1. **Problem Statement:** Find the HCF of 408 and 1023 expressed in the form $1023x + 408y$ and determine the values of $x$ and $y$. 2. **Formula and Method:** The HCF (Highest Common Factor) of two numbers can be expressed as a linear combination of those numbers using the Extended Euclidean Algorithm: $$\text{HCF}(a,b) = ax + by$$ where $x$ and $y$ are integers. 3. **Step-by-step solution:** - Apply the Euclidean Algorithm to find HCF(1023, 408): $$1023 = 408 \times 2 + 207$$ $$408 = 207 \times 1 + 201$$ $$207 = 201 \times 1 + 6$$ $$201 = 6 \times 33 + 3$$ $$6 = 3 \times 2 + 0$$ So, HCF is 3. - Now express 3 as a combination of 1023 and 408 by back substitution: $$3 = 201 - 6 \times 33$$ $$6 = 207 - 201 \times 1$$ Substitute $6$: $$3 = 201 - (207 - 201) \times 33 = 201 \times 34 - 207 \times 33$$ Substitute $201 = 408 - 207$: $$3 = (408 - 207) \times 34 - 207 \times 33 = 408 \times 34 - 207 \times 67$$ Substitute $207 = 1023 - 408 \times 2$: $$3 = 408 \times 34 - (1023 - 408 \times 2) \times 67 = 408 \times 34 - 1023 \times 67 + 408 \times 134$$ Combine terms: $$3 = 408 \times (34 + 134) - 1023 \times 67 = 408 \times 168 - 1023 \times 67$$ Thus, $$3 = 1023 \times (-67) + 408 \times 168$$ 4. **Answer:** The HCF of 408 and 1023 is 3, and it can be expressed as: $$3 = 1023 \times (-67) + 408 \times 168$$ Hence, $x = -67$ and $y = 168$.