Let's solve part (a) step by step, as requested: ## (a) d = GCD(a = 1735, b = 24) and the (x, y) where d = ax + by = gcd(a, b) --- ### **Step 1: Find GCD using the Euclidean Algorithm** We apply the Euclidean algorithm: - \( a = 1735 \) - \( b = 24 \) Calculate remainders: 1. \( 1735 \div 24 = 72 \) remainder \( 7 \) \( 1735 = 24 \times 72 + 7 \) 2. \( 24 \div 7 = 3 \) remainder \( 3 \) \( 24 = 7 \times 3 + 3 \) 3. \( 7 \div 3 = 2 \) remainder \( 1 \) \( 7 = 3 \times 2 + 1 \) 4. \( 3 \div 1 = 3 \) remainder \( 0 \) \( 3 = 1 \times 3 + 0 \) **Explanation:** The last non-zero remainder is the GCD. Thus, **GCD(1735, 24) = 1**. --- ### **Step 2: Use Extended Euclidean Algorithm to find x and y** Now, backtrack to express 1 as a linear combination: From above: - \( 1 = 7 - 2 \times 3 \) - But \( 3 = 24 - 3 \times 7 \), so: \( 1 = 7 - 2 \times (24 - 3 \times 7) \) \( 1 = 7 - 2 \times 24 + 6 \times 7 \) \( 1 = 7 + 6 \times 7 - 2 \times 24 \) \( 1 = 7 \times 7 - 2 \times 24 \) But \( 7 = 1735 - 72 \times 24 \): \( 1 = (1735 - 72 \times 24) \times 7 - 2 \times 24 \) \( 1 = 1735 \times 7 - 72 \times 24 \times 7 - 2 \times 24 \) \( 1 = 1735 \times 7 - (72 \times 7 + 2) \times 24 \) \( 1 = 1735 \times 7 - 506 \times 24 \) So, **x = 7, y = -506** **Explanation:** We found integers x and y such that \( 1735x + 24y = 1 \). --- ## **Final Answer:** - **GCD(1735, 24) = 1** - **x = 7, y = -506** - **Are they relatively prime?** **Yes, since the GCD is 1.** --- If you want steps for (b) and (c), let me know!

Certainly! Based on the previous question, here's a structured two-step solution with explanations and the final answer. --- ### **Part (a): Find GCD(1735, 24) and coefficients x, y such that \(1735x + 24y = \gcd(1735, 24)\)** --- ### **Step 1: Compute GCD using Euclidean Algorithm** **Calculation:** - \(1735 \div 24 = 72\) remainder **7** - \(24 \div 7 = 3\) remainder **3** - \(7 \div 3 = 2\) remainder **1** - \(3 \div 1 = 3\) remainder **0** **Explanation:** The last non-zero remainder is **1**, so \[ \boxed{\gcd(1735, 24) = 1} \] --- ### **Step 2: Use Extended Euclidean Algorithm to find x and y** Back-substitute to express 1 as a combination of 1735 and 24: - From \(7 = 1735 - 72 \times 24\) - From \(3 = 24 - 3 \times 7\) - From \(1 = 7 - 2 \times 3\) Substitute \(3\): \[ 1 = 7 - 2 (24 - 3 \times 7) = 7 - 2 \times 24 + 6 \times 7 = 7 + 6 \times 7 - 2 \times 24 = 7 \times 7 - 2 \times 24 \] Now, substitute \(7 = 1735 - 72 \times 24\): \[ 1 = (1735 - 72 \times 24) \times 7 - 2 \times 24 = 1735 \times 7 - 72 \times 24 \times 7 - 2 \times 24 \] Simplify: \[ 1 = 1735 \times 7 - 504 \times 24 - 2 \times 24 = 1735 \times 7 - (504 + 2) \times 24 = 1735 \times 7 - 506 \times 24 \] **Therefore:** \[ \boxed{ x = 7,\quad y = -506 } \] --- ### **Final Answer:** - \(\boxed{\gcd(1735, 24) = 1}\) - \(\boxed{1735 \times 7 + 24 \times (-506) = 1}\) Yes, they are coprime, and the coefficients are as above. --- Let me know if you'd like solutions for parts (b) and (c)!

Certainly! Here is the step-by-step solution for part (b), following the same two-step approach with explanations and the final answer: --- ### **Part (b): Find integers \(a, b, c\) such that \(d = \gcd(a, b) = 357.5\) and the linear combination \(a x + b y = d\)** (Note: Since \(\gcd\) must be an integer, and 357.5 is not an integer, this suggests a possible typo or misinterpretation. Typically, gcd values are integers. Assuming the intended value is 357 or 358, but since the problem states 357.5, perhaps it's a different context. If it is a real number, gcd isn't defined in the usual integer sense. Could you clarify? For the sake of illustration, I'll proceed assuming the gcd is 357.) --- ### **Step 1: Verify if \(d = 357\) divides \(a\) and \(b\)** Suppose: - \(a = 357.5\) (but this isn't an integer, so gcd isn't defined in integers) - Alternatively, if the number was 357, then: If \(a = 357\), \(b = 24\): Calculate \(\gcd(357, 24)\): - \(357 \div 24 = 14\) remainder \(21\) - \(24 \div 21 = 1\) remainder \(3\) - \(21 \div 3 = 7\) remainder \(0\) **GCD**: \[ \boxed{\gcd(357, 24) = 3} \] **Explanation:** The gcd of 357 and 24 is 3, which divides both numbers. --- ### **Step 2: Find integers \(x, y\) such that \(357 x + 24 y = 3\)** Using the Extended Euclidean Algorithm: From previous step: - \(357 = 24 \times 14 + 21\) - \(24 = 21 \times 1 + 3\) - \(21 = 3 \times 7 + 0\) Back-substitute: 1. \(3 = 24 - 21 \times 1\) But \(21 = 357 - 24 \times 14\), so: \[ 3 = 24 - (357 - 24 \times 14) \times 1 = 24 - 357 + 24 \times 14 = 24 \times 15 - 357 \] Expressed as: \[ 3 = (-1) \times 357 + 15 \times 24 \] **Answer:** \[ x = -1,\quad y = 15 \] **Final notes:** - The gcd is 3. - The coefficients satisfy \(357 \times (-1) + 24 \times 15 = 3\). --- ### **Final Result for part (b):** \[ \boxed{ \text{gcd}(357, 24) = 3, \quad \text{and} \quad -1 \times 357 + 15 \times 24 = 3 } \] --- **Please clarify if the initial value 357.5 was intentional or if it was a typo.**

Certainly! Here's the structured two-step solution for part (c), based on the pattern of previous parts: --- ### **Part (c): Find integers \(a, b, c\) such that \(d = \gcd(a, b) = 17.101\) and the linear combination \(a x + b y = d\)** **Note:** Since \(\gcd\) is only defined for integers, and 17.101 is a decimal, it suggests a possible typo or misinterpretation. Typically, gcd values are integers. Assuming the intended value is 17, here is the solution: --- ### **Step 1: Calculate \(\gcd(17, 101)\)** Using Euclidean Algorithm: - \(101 \div 17 = 5\) remainder \(16\) - \(17 \div 16 = 1\) remainder \(1\) - \(16 \div 1 = 16\) remainder \(0\) **GCD:** \[ \boxed{\gcd(17, 101) = 1} \] **Explanation:** The gcd of 17 and 101 is 1, indicating they are coprime. --- ### **Step 2: Use Extended Euclidean Algorithm to find \(x, y\) such that \(17 x + 101 y = 1\)** Backtracking: - \(1 = 17 - 16 \times 1\) From earlier: - \(16 = 101 - 17 \times 5\) Substitute: \[ 1 = 17 - (101 - 17 \times 5) \times 1 = 17 - 101 + 17 \times 5 = 17 \times 6 - 101 \] Expressed as: \[ 1 = 6 \times 17 - 1 \times 101 \] **Coefficients:** \[ x = 6,\quad y = -1 \] --- ### **Final Answer:** - \(\gcd(17, 101) = 1\) - **Linear combination:** \(17 \times 6 + 101 \times (-1) = 1\) **Summary:** They are coprime, and the coefficients \(x=6\), \(y=-1\) satisfy the linear combination. --- **If the decimal values (17.101) were intentional for a different context, please clarify.**