Question

The greatest common divisor \((G C D)\) of two integers is the largest integer that evenly divides each of the numbers. Write a function gcd that returns the greatest common divisor of two integers.

Short answer

Use the Euclidean algorithm in a function to find the GCD of two integers.

Step-by-step solution

Understand the Task

We need to write a function to calculate the greatest common divisor (GCD) of two integers. The GCD is the largest positive integer that divides both numbers without leaving a remainder.

Choose an Algorithm

The Euclidean algorithm is efficient for finding the GCD. It uses the principle that the GCD of two numbers also divides their difference. We will implement this algorithm to solve the problem.

Implement the Euclidean Algorithm

We will create a function called `gcd` that takes two integers `a` and `b` and returns their GCD using the following steps: 1. If `b` is 0, return `a` as the GCD. 2. Otherwise, set `a` to `b` and `b` to the remainder of the division `a % b`, and repeat the process.

Write the Function

Here's how the function is written in Python: ```python def gcd(a, b): while b != 0: a, b = b, a % b return a ```

Test the Function

To ensure the function works correctly, test it with some example inputs: - `gcd(48, 18)` should return 6. - `gcd(101, 103)` should return 1, since these numbers are coprime. - `gcd(0, 5)` should return 5, since any number is a divisor of 0.

Key concepts

Understanding the Euclidean Algorithm

The Euclidean algorithm is a well-known and efficient method to compute the greatest common divisor (GCD) of two positive integers. This algorithm is based on a few simple principles that make it both powerful and straightforward. At its core, the Euclidean algorithm relies on the fact that the GCD of two numbers is the same as the GCD of the smaller number and the remainder of dividing the larger number by the smaller one.

Here's how the algorithm typically works:

• Take two numbers, say `a` and `b`, where `a > b`.
• Calculate the remainder of `a` divided by `b`, that is `r = a % b`.
• Replace `a` with `b`, and `b` with `r`.
• Repeat the steps until `b` becomes 0.

When `b` is zero, the GCD is the current value of `a`. This process is fast and often involves a few iterations, especially for smaller numbers. Additionally, it doesn't require any form of multiplication or division beyond finding the remainder, which makes it computationally efficient.

Greatest Common Divisor (GCD) Defined

The greatest common divisor (GCD) of two integers is the largest integer that can exactly divide both numbers without leaving a remainder. Finding the GCD is a common task in mathematics and computer science because it simplifies problems related to ratios and fractions.

Consider the numbers 48 and 18 as an example:

• The divisors of 48 include: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
• The divisors of 18 include: 1, 2, 3, 6, 9, 18.
• The largest common divisor from these lists is 6, making it the GCD of 48 and 18.

Finding the GCD is especially useful in reducing fractions to their lowest terms and when determining common denominators. The Euclidean algorithm simplifies this process greatly by providing an iterative method to find the GCD without listing all divisors.

Implementing a GCD Function in C++

To solve the problem of finding the GCD using C++, we apply the logic of the Euclidean algorithm in a function. Implementing a function involves defining the function parameters and using a loop to follow the algorithm's steps until the desired result is obtained.

Below is a simple implementation of the GCD function in C++: ```cpp int gcd(int a, int b) { while (b != 0) { int temp = b; b = a % b; a = temp; } return a; } ``` This function works by continuing to apply the algorithm until `b` becomes zero, at which point `a` contains the GCD. Let's break down this code:

• The function `gcd` takes two integer arguments, `a` and `b`.
• Within the `while` loop, the remainder of `a` divided by `b` replaces `b`, and `a` takes the value of `b`.
• The loop continues until `b` equals zero, ensuring that `a` is the GCD at the end.

This function is efficient and concise, making it suitable for various applications where computation of the GCD is necessary.