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Chapter 10: Problem 14

Illustrate the flow of Algorithm 10.1 when the top-level call is gcd (68,40).

Short Answer

Expert verified
The gcd of 68 and 40 is 4. The algorithm was successful in finding the greatest common divisor using Euclid's Algorithm.

Step by step solution

01

Understand the Algorithm

The gcd function is a common function used in programming and math which finds the greatest common divisor of two integers. In Algorithm 10.1, it uses a recursive approach based on Euclid's Algorithm. The Euclidean Algorithm states that the greatest common divisor of two numbers \(a\) and \(b\) can be obtained by computing the greatest common divisor of \(b\) and \(a \% b\). This process is repeated until \(b\) equals to 0. At that point, \(a\) is the result.
02

Initial Input

We start with gcd(68, 40). Here, \(a = 68\) and \(b = 40\). Since \(b\) is not equal to 0, we continue the algorithm running the gcd function again, but this time with \(b\) being 40 and \(a \% b\) being 68 modulo 40, which is 28. So, the next step is to calculate gcd(40, 28).
03

Continue Recursion

Now, we compute gcd(40, 28), where \(a = 40\) and \(b = 28\). Again, \(b\) isn’t 0, so we continue with the algorithm: gcd(28, 40 % 28) becomes gcd(28, 12).
04

Continue Until \(b\) is 0

We then repeat the steps. Compute gcd(28, 12), which results in gcd(12, 28 % 12) which is gcd(12, 4). The next step is gcd(12, 4), resulting in gcd(4,12 % 4) which is gcd(4, 0). At this point \(b\) is equal to 0 and per Euclidean algorithm, we stop and the current \(a\) is the result.

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