Absolutely! Let’s follow your instructions step-by-step for **Question 4** from the image (Euclidean algorithm for GCD): --- ### **Step 1: Introduction of the Concept** The **Euclidean algorithm** is a classic method to compute the greatest common divisor (gcd) of two positive integers, \(a\) and \(b\) (where \(a > b\)). The gcd is the largest integer that divides both numbers without leaving a remainder. There are two main versions of this algorithm: iterative and recursive. --- ### **Step 2: Part (a) - Iterative Version of the Euclidean Algorithm** **Algorithm (Iterative):** 1. While \(b \neq 0\): - Set \(temp = b\) - Set \(b = a \mod b\) - Set \(a = temp\) 2. When \(b = 0\), \(a\) is the gcd. **Explanation:** We repeatedly replace \(a\) with \(b\) and \(b\) with the remainder of \(a\) divided by \(b\), until \(b\) becomes zero. At this point, \(a\) is the greatest common divisor. --- ### **Step 3: Part (b) - Recursive Version of the Euclidean Algorithm** **Algorithm (Recursive):** - If \(b = 0\), return \(a\). - Otherwise, return \(gcd(b, a \mod b)\). **Explanation:** This version calls itself with the next pair \((b, a \mod b)\) until \(b\) becomes zero, at which point \(a\) is the gcd. --- ### **Step 4: Final Answer in Very Few Sentences** The Euclidean algorithm efficiently finds the gcd of two numbers using either iteration or recursion by repeatedly applying the operation \(gcd(a, b) = gcd(b, a \mod b)\) until the second number becomes zero. --- ### **Step 5: Explanation of What We Did in Each Step** - **Step 1:** Introduced the concept of finding gcd using the Euclidean algorithm. - **Step 2:** Explained the iterative approach with steps. - **Step 3:** Provided the recursive approach and its logic. - **Step 4:** Summarized both methods in one concise answer. --- Let me know if you want the solution for another question or a sample code!