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Chapter 1: Problem 5

Write an algorithm that finds the greatest common divisor of two integers.

Short Answer

Expert verified
A simple algorithm using Euclid's method is used to find the Greatest Common Divisor (GCD) of two integers. The method replaces the two integers with the second integer and the mod of the first integer by the second until the second integer reaches 0. The GCD is then the remaining non-zero integer.

Step by step solution

01

Understand Euclid’s Algorithm

Euclid’s Algorithm states that the greatest common divisor of two numbers \(a\) and \(b\) is the same as the greatest common divisor of \(b\) and \(a mod b\). In other words, if we replace \(a\) and \(b\) with \(b\) and \(a mod b\) respectively, and repeat this process till \(b\) becomes \(0\), we are left with the GCD.
02

Declare the Variables

Declare two integer type variables \(a\) and \(b\). These will hold the two numbers for which we are to find the GCD.
03

Implement Euclid's Algorithm

Implement a loop that continues until \(b\) does not equal to \(0\). In this loop, set \(a\) to \(b\) and \(b\) to \(a mod b\).
04

Return the GCD

Once \(b\) reaches \(0\), break the loop and return \(a\) as the GCD of the original two numbers.

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Most popular questions from this chapter

Show the correctness of the following statements. a. \(\lg n \in O(n)\) b. \(n \in O(n \lg n)\) c. \(n \lg n \in O\left(n^{2}\right)\) d. \(2^{n} \in \Omega\left(5^{\ln n}\right.\) e. \(\lg ^{3} n \in o\left(n^{0.5}\right)\)

Using the definitions of \(O\) and \(\Omega\), show that $$6 n^{2}+20 n \in O\left(n^{3}\right) \quad \text { but } \quad 6 n^{2}+20 n \notin \Omega\left(n^{3}\right)$$

Let \(p(n)=a_{k} n^{k}+a_{k-1} n^{k-1}+\ldots+a_{1} n+a_{0},\) where \(a_{k}>0 .\) Using the Properties or order in Section \(1.4 .2,\) show that \(p(n) \in \Theta\left(n^{k}\right)\)

Give a \(\Theta(n \text { lg } n)\) algorithm that computes the reminder when \(x^{n}\) is divided by \(p .\) For simplicity, you may assume that \(n\) is a power of 2 That is, \(n=2^{k}\) for some positive integer \(k\)

Write a \(\Theta(n)\) algorithm that sorts \(n\) distinct integers, ranging in size between 1 and \(k n\) inclusive, where \(k\) is a constant positive integer. (Hint: Use a kn-element array.)
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