Question

Write a method is Multiple that determines, for a pair of integers, whether the second integer is a multiple of the first. The method should take two integer arguments and return true if the second is a multiple of the first and false otherwise. [Hint: Use the remainder operator.] Incorporate this method into an application that inputs a series of pairs of integers (one pair at a time) and determines whether the second value in each pair is a multiple of the first.

Short answer

Create a method isMultiple that returns true if the second integer is a multiple of the first (i.e., secondInt % firstInt == 0), and false otherwise. Use this method in an application which takes pairs of integers from the user, checks the multiple relationship using isMultiple, and informs the user of the result.

Step-by-step solution

Understand the Remainder Operator

The remainder operator, denoted by % in many programming languages, gives the remainder after dividing the first number by the second. In the context of this exercise, an integer A is a multiple of another integer B if A % B equals 0.

Define isMultiple Method

Create a method named isMultiple with two integer parameters. The method should use the remainder operator to check if the second integer is a multiple of the first by returning the result of the expression secondInt % firstInt == 0.

Incorporate isMultiple into an Application

Write a main application function or routine where users can input pairs of integers. After inputting each pair, call the isMultiple method with those integers and report to the user whether the second integer is a multiple of the first, based on the returned value from the isMultiple method.

Test the Application

Test the application by inputting pairs of integers to see if the isMultiple method is correctly determining the relationship between the numbers. Verify the accuracy with known multiples and non-multiples to ensure the application behaves as expected.

Key concepts

isMultiple Method

The 'isMultiple' method is a simple yet powerful tool for determining the divisibility of two integers. The essence of this method lies in its ability to answer whether an integer 'B' is a multiple of another integer 'A'. To declare this method in Java, you'd typically create a function that takes two integer parameters. Within the function, a crucial operator comes into play - the remainder operator. Applying this operator involves calculating the remainder of dividing 'B' by 'A'.

If the result is zero, 'B' can be considered a multiple of 'A' since it divides evenly with no remainder left. Hence, the method would return 'true' in such a case. Otherwise, it returns 'false', indicating that 'B' does not constitute a whole number of 'A's. To make this concept crystal clear to students, consider using examples with known multiples and non-multiples when explaining the 'isMultiple' method implementation.

Integer Division

In Java, and indeed most programming languages, when you divide one integer by another using the '/' operator, the result is an integer division. This might be somewhat surprising if you're expecting the precision of real numbers in the result. What actually happens is that the fractional part of the result is discarded, and you're left with just the whole number.

This process is akin to asking, 'If I have 'B' apples and want to make bags containing 'A' apples each, how many full bags can I make?' The answer would ignore any partial bags. Understanding integer division helps in grasping the underlying mechanism of the remainder operator since they are closely related: integer division gives you the number of full 'bags', while the remainder operator would tell you about any 'leftover apples'.

Modular Arithmetic in Programming

Modular arithmetic, popularly known as 'clock arithmetic', is deeply interwoven into programming, through the remainder operator - often symbolized as '%'. This form of arithmetic involves a wrap-around feature; for instance, in a 12-hour clock system, one hour past 12 o'clock takes us back to 1 o'clock.

In programming, we utilize this concept using the remainder operator to easily determine the residue of a division operation. It has a multitude of applications, such as cyclic data structures, cryptography, and algorithms that need to loop within a fixed set of values. It's an elegant way to ensure that values stay within a certain range, by naturally looping over once they hit a specific limit. This concept is pivotal in understanding scenarios where you need to check for divisibility or iterate over a fixed range of values, such as determining if a number is a multiple of another as seen in the 'isMultiple' method.

Java Method Implementation

When implementing methods in Java, clarity and reusability should be at the forefront. A method in Java essentially encapsulates a sequence of statements that perform a specific task, which can be invoked with specified parameters to execute its logic. For the 'isMultiple' method, it's about succinctly checking whether one integer is a multiple of another.

The methodology is straightforward: define the method signature with appropriate parameter names and types, and then code the logic inside the method body. In keeping with good coding practices, the method should only perform this single task and do it well. Testing is just as crucial as implementation; therefore, ensure to vet the method against varied input pairs. Properly implemented and tested Java methods are like well-oiled cogs in the machine of a program, allowing for modular, understandable, and maintainable code.