Question

The following is Euclid's 2,300-year-old algorithm for finding the greatest common divisor of two positive integers \(/\) and \(J\). Step Operation 1 Get two positive integers as input; call the larger value / and the smaller value J 2 Divide \(I\) by \(J\), and call the remainder \(R\) 3 If \(R\) is not 0 , then reset \(/\) to the value of \(J\), reset \(J\) to the value of \(R\), and go back to Step 2 4 Print out the answer, which is the value of \(J\) 5 Stop a. Go through this algorithm using the input values 20 and 32 . After each step of the algorithm is completed, give the values of \(I\), \(J\), and \(R\). Determine the final output of the algorithm. b. Does the algorithm work correctly when the two inputs are 0 and 32 ? Describe exactly what happens, and modify the algorithm so that it gives an appropriate error message.

Short answer

The GCD of 20 and 32 is 4. The modified algorithm handles the zero case with an error message.

Step-by-step solution

Initial Values

Start with the numbers 20 and 32. According to the algorithm, the larger value is assigned to \(I\) and the smaller to \(J\). Thus, \(I = 32\) and \(J = 20\).

Calculate Remainder

Divide \(I\) by \(J\) to find the remainder \(R\). For \(I = 32\) and \(J = 20\), \(R = 32 \mod 20 = 12\).

Reset Values

Since \(R eq 0\), set \(I = J = 20\) and \(J = R = 12\).

Repeat Division

Divide \(I = 20\) by \(J = 12\), resulting in \(R = 20 \mod 12 = 8\).

Update Values Again

Since the remainder \(R = 8\) is not zero, set \(I = J = 12\) and \(J = R = 8\).

Continue Algorithm

Divide \(I = 12\) by \(J = 8\), giving \(R = 12 \mod 8 = 4\).

Further Update

Since \(R = 4\) is not zero, reassign \(I = J = 8\) and \(J = R = 4\).

Yet Another Division

Now, divide \(I = 8\) by \(J = 4\), and find \(R = 8 \mod 4 = 0\).

Check Termination Condition

Since \(R = 0\), stop the algorithm and print the current value of \(J\), which is 4.

Check Special Case (0 and 32)

Attempt to run the algorithm with \(I = 32\) and \(J = 0\). Division by zero is undefined and would cause an error.

Modify Algorithm

To handle the case where one of the inputs is zero, check at the beginning if \(J = 0\). If \(J = 0\), print an error message: "Error: One of the inputs is zero."

Key concepts

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest integer that divides both numbers without leaving a remainder. In simpler terms, it's the biggest number that fits into both numbers perfectly. For instance, the GCD of 8 and 12 is 4, because 4 is the largest number that divides both 8 and 12 without any leftovers.
Understanding the GCD is essential in various aspects of mathematics, including simplifying fractions and solving problems involving ratios. It's a foundational concept that helps in reducing numbers into their simplest forms.

• To find the GCD, you can use multiple methods like listing factors, using prime factorization, or applying the Euclidean algorithm.
• The Euclidean algorithm is particularly efficient for larger numbers because it reduces the problem step by step until the answer is clear.

This foundational concept will lead us into understanding how one of the most efficient ways to compute the GCD is through the Euclidean algorithm.

Algorithm Steps

The Euclidean Algorithm is a step-by-step process to calculate the GCD of two numbers. The beauty of this algorithm is its efficiency and simplicity. Here's a breakdown of the steps involved:

• Step 1: Begin with two positive integers, and assign the larger value to \(I\) and the smaller to \(J\).
• Step 2: Divide \(I\) by \(J\) and find the remainder \(R\).
• Step 3: If \(R\) is not zero, update \(I\) with \(J\), and \(J\) with \(R\), then repeat from Step 2.
• Step 4: When \(R = 0\), the current value of \(J\) is the GCD. Output this value.
• Step 5: Terminate the algorithm.

Through repetition and reduction, the algorithm efficiently narrows down the possible values until the greatest common divisor is identified. This process ensures that you have an answer quickly even for large numbers. Understanding each step thoroughly can significantly enhance your comprehension of problem-solving in mathematics.

Error Handling

While the Euclidean Algorithm is efficient for computing the GCD of two positive integers, certain edge cases need special attention, particularly when one of the input numbers is zero. In mathematical terms, dividing by zero is undefined and can cause errors during computation.
To handle such errors, it's crucial to include a check at the start of the algorithm. Here’s how you can manage these exceptions:

• Start the algorithm by verifying if either number is zero.
• If one of the numbers is zero, you should provide an appropriate error message or simply output the non-zero number as the GCD (since any non-zero number is divisible by zero). For example: "Error: One of the inputs is zero."
• Alternatively, if one number is zero and the other is valid, the valid number is the GCD because it is the only non-zero divisor available.
• Make sure to apply this check before performing any divisions to prevent computational errors and provide meaningful results.

By addressing these potential errors, you ensure the algorithm is robust and can handle any input, maintaining accuracy and reliability in your calculations.