SHOW ALL YOUR WORK. REMEMBER THAT PROGRAM SEGMENTS ARE TO BE WRITTEN IN JAVA.

Assume that the classes listed in the Java Quick Reference have been imported where appropriate.
Unless otherwise noted in the question, assume that parameters in method calls are not null and that methods are called only when their preconditions are satisfied.
In writing solutions for each question, you may use any of the accessible methods that are listed in classes defined in that question. Writing significant amounts of code that can be replaced by a call to one of these methods will not receive full credit.
 
This question involves computing the greatest common factor between two positive integers and using greatest common factors to reduce fractions. You will write two methods in the NumberSystem class that follows.

public class NumberSystem

{

/\*\* Precondition: a and b are positive integers.

\* Returns the greatest common factor of a and b, as described in part (a).

\*/

public static int gcf(int a, int b)

{ /\* to be implemented in part (a) \*/ }

 

/\*\* Precondition: numerator and denominator are positive integers.

\* Reduces the fraction numerator / denominator

\* and prints the result, as described in part (b).

\*/

public static void reduceFraction(int numerator, int denominator)

{ /\* to be implemented in part (b) \*/ }

}

The greatest common factor (GCF) of two integers a and b is the largest integer that divides evenly into both a and b. For example, the GCF of 8 and 12 is 4.

The greatest common factor can be computed using the following rules.

Case I: If a is evenly divisible by b, then the GCF is b.
Case II: If a is not evenly divisible by b, then the GCF of a and b is equal to the GCF of b and the remainder when a is divided by b.
If the rule in case II is repeatedly applied, it is guaranteed to eventually result in case I. Consider the following examples.

Example 1
In determining the GCF of 30 and 3, case I applies because 30 is evenly divisible by 3. Therefore, the GCF of 30 and 3 is 3.
Example 2
In determining the GCF of 3 and 30, case II applies because 3 is not evenly divisible by 30. The GCF of 3 and 30 will be equal to the GCF of 30 and the remainder when 3 is divided by 30, or 3.
In determining the GCF of 30 and 3, case I applies because 30 is evenly divisible by 3. The GCF of 30 and 3 is 3, and therefore the GCF of 3 and 30 is also 3.
Example 3
In determining the GCF of 24 and 9, case II applies because 24 is not evenly divisible by 9. The GCF of 24 and 9 will be equal to the GCF of 9 and the remainder when 24 is divided by 9, or 6.
In determining the GCF of 9 and 6, case II applies because 9 is not evenly divisible by 6. The GCF of 9 and 6 will be equal to the GCF of 6 and the remainder when 9 is divided by 6, or 3.
In determining the GCF of 6 and 3, case I applies because 6 is evenly divisible by 3. The GCF of 6 and 3 is 3, and therefore the GCF of 24 and 9 is also 3.
Example 4
In determining the GCF of 7 and 3, case II applies because 7 is not evenly divisible by 3. The GCF of 7 and 3 will be equal to the GCF of 3 and the remainder when 7 is divided by 3, or 1.
In determining the GCF of 3 and 1, case I applies because 3 is evenly divisible by 1. The GCF of 3 and 1 is 1, and therefore the GCF of 7 and 3 is also 1.
 

(a) The gcf method returns the greatest common factor of parameters a and b, as determined by case I and case II. Write the gcf method below. You are encouraged to implement this method recursively.

/\*\* Precondition: a and b are positive integers.

\* Returns the greatest common factor of a and b, as described in part (a).

\*/

public static int gcf(int a, int b)
user
Sure! I'll help you with the implementation. Here's how you can write the `gcf` method recursively based on the rules you've provided:

```java
/**
 * Precondition: a and b are positive integers. * Returns the greatest common factor of a and b, as described in part (a). */
public static int gcf(int a, int b) {
    if (b == 0) { // Base case: if b is 0, a is the GCF
        return a;
    } else {
        // Recursive case: apply case II until case I
        return gcf(b, a % b);
    }
}
```

This method works as follows:

- If `b` is 0, which means `a` is evenly divisible by `b`, then `a` is the GCF. We return `a` in this case.
- Otherwise, we call `gcf` recursively with `b` and the remainder when `a` is divided by `b`. The modulus operator `%` gives us this remainder. This way, we're ensuring we eventually end up in the base case (when `b` is 0).

As for the `reduceFraction` method, it should be relatively simple. Here's how you might write it:

```java
/**
 * Precondition: numerator and denominator are positive integers. * Reduces the fraction numerator / denominator and prints the result. */
public static void reduceFraction(int numerator, int denominator) {
    int gcd = gcf(numerator, denominator); // Compute GCF using gcf method
    if (gcd != 1) { // If GCF is not 1, reduce fraction
        int quotient = numerator / gcd;
        int remainder = numerator % gcd;
        numerator = quotient * denominator + remainder;
        denominator = gcd;
        System.out.println("Reduced fraction: " + numerator + "/" + denominator);
    } else {
        System.out.println("Fraction is already in simplest form.");
    }
}
```

This method first computes the GCF using the `gcf` method we previously defined. If the GCF is not 1 (i.e. The fraction has a non-trivial GCF), it reduces the fraction using the Euclidean algorithm. Otherwise, it prints a message indicating that the fraction is already in simplest form.

Note: You may also want to modify this method to handle negative numbers and fractions with non-integer numerators or denominators.
assistant
SHOW ALL YOUR WORK. REMEMBER THAT PROGRAM SEGMENTS ARE TO BE WRITTEN IN JAVA.

Assume that the classes listed in the Java Quick Reference have been imported where appropriate.
Unless otherwise noted in the question, assume that parameters in method calls are not null and that methods are called only when their preconditions are satisfied.
In writing solutions for each question, you may use any of the accessible methods that are listed in classes defined in that question. Writing significant amounts of code that can be replaced by a call to one of these methods will not receive full credit.
 
This question involves computing the greatest common factor between two positive integers and using greatest common factors to reduce fractions. You will write two methods in the NumberSystem class that follows.

public class NumberSystem

{

/\*\* Precondition: a and b are positive integers.

\* Returns the greatest common factor of a and b, as described in part (a).

\*/

public static int gcf(int a, int b)

{ /\* to be implemented in part (a) \*/ }

 

/\*\* Precondition: numerator and denominator are positive integers.

\* Reduces the fraction numerator / denominator

\* and prints the result, as described in part (b).

\*/

public static void reduceFraction(int numerator, int denominator)

{ /\* to be implemented in part (b) \*/ }

}

The greatest common factor (GCF) of two integers a and b is the largest integer that divides evenly into both a and b. For example, the GCF of 8 and 12 is 4.

The greatest common factor can be computed using the following rules.

Case I: If a is evenly divisible by b, then the GCF is b.
Case II: If a is not evenly divisible by b, then the GCF of a and b is equal to the GCF of b and the remainder when a is divided by b.
If the rule in case II is repeatedly applied, it is guaranteed to eventually result in case I. Consider the following examples.

Example 1
In determining the GCF of 30 and 3, case I applies because 30 is evenly divisible by 3. Therefore, the GCF of 30 and 3 is 3.
Example 2
In determining the GCF of 3 and 30, case II applies because 3 is not evenly divisible by 30. The GCF of 3 and 30 will be equal to the GCF of 30 and the remainder when 3 is divided by 30, or 3.
In determining the GCF of 30 and 3, case I applies because 30 is evenly divisible by 3. The GCF of 30 and 3 is 3, and therefore the GCF of 3 and 30 is also 3.
Example 3
In determining the GCF of 24 and 9, case II applies because 24 is not evenly divisible by 9. The GCF of 24 and 9 will be equal to the GCF of 9 and the remainder when 24 is divided by 9, or 6.
In determining the GCF of 9 and 6, case II applies because 9 is not evenly divisible by 6. The GCF of 9 and 6 will be equal to the GCF of 6 and the remainder when 9 is divided by 6, or 3.
In determining the GCF of 6 and 3, case I applies because 6 is evenly divisible by 3. The GCF of 6 and 3 is 3, and therefore the GCF of 24 and 9 is also 3.
Example 4
In determining the GCF of 7 and 3, case II applies because 7 is not evenly divisible by 3. The GCF of 7 and 3 will be equal to the GCF of 3 and the remainder when 7 is divided by 3, or 1.
In determining the GCF of 3 and 1, case I applies because 3 is evenly divisible by 1. The GCF of 3 and 1 is 1, and therefore the GCF of 7 and 3 is also 1.
 

(a) The gcf method returns the greatest common factor of parameters a and b, as determined by case I and case II. Write the gcf method below. You are encouraged to implement this method recursively.

/\*\* Precondition: a and b are positive integers.

\* Returns the greatest common factor of a and b, as described in part (a).

\*/

public static int gcf(int a, int b)
user
Sure, I can help you write the `gcf` method in Java. As suggested, we'll use a recursive approach based on the rules you provided. Here's how you can do it:

```java
public static int gcf(int a, int b)
{
    if (b == 0) {
        return a;
    } else {
        return gcf(b, a % b);
    }
}
```

In this method, we first check if `b` is zero. If it is, then `a` is the greatest common factor (case I). If `b` is not zero, we recursively call the `gcf` method with `b` and the remainder of `a` divided by `b` (case II). This process continues until we reach a point where `b` is zero, at which point we return `a` as the greatest common factor.
assistant