Question

Write a function multiple that determines for a pair of integers whether the second is a multiple of the first. The function should take two integer arguments and return true if the second is a multiple of the first, false otherwise. Use this function in a program that inputs a series of pairs of integers.

Short answer

Implement a function `multiple(a, b)` that checks `b % a == 0`. Use a loop to input pairs and print whether they are multiples.

Step-by-step solution

Understand the Concept of Multiples

A number b is a multiple of a number a if there exists an integer n such that a * n = b. In other words, b is evenly divisible by a, and the remainder of the division is zero.

Define the Signature of the Function

We need to write a function named `multiple` that takes two integer arguments. This function will return a boolean value.

Implement the Multiple Function

In the function `multiple(a, b)`, check if b % a == 0. If true, return `True`, else return `False`. This can be written in Python as follows: ```python def multiple(a, b): return b % a == 0 ```

Develop a Program to Use the Function

Create a program that takes a series of integer pairs as input, calls the `multiple` function for each pair, and outputs the result. This can be done using a loop to repeatedly ask for input until the user decides to stop.

Implement the Program Logic

Here's an example program in Python: ```python while True: # Input the integers a = int(input("Enter the first integer: ")) b = int(input("Enter the second integer: ")) # Use the function and print result if multiple(a, b): print(f"{b} is a multiple of {a}.") else: print(f"{b} is not a multiple of {a}.") # Prompt for continuation cont = input("Do you want to check another pair of numbers? (yes/no): ") if cont.lower() != 'yes': break ```

Key concepts

Integers in Programming

Integers are fundamental data types in programming, representing whole numbers without a fractional component. They can be positive, negative, or zero. In programming, integers are used for counting, indexing, and performing arithmetic operations.

• **Data Representation**: Integers in programming are often represented as binary numbers. The size of the integer (such as 32-bit or 64-bit) can determine the range of values it can hold.
• **Operations**: Basic operations like addition, subtraction, multiplication, and division are performed using integers. Additionally, the modulus operation (%) is crucial as it returns the remainder of a division, which is helpful in determining properties like multiples.

In many programming languages, integers provide a straightforward way to handle numbers for processing and logic building. Consider this exercise where the problem involves checking if one integer is a multiple of another. This uses the modulus operation to find if dividing one integer by another leaves no remainder.

Boolean Logic

Boolean logic is a branch of algebra in which variables have true or false values. It's crucial in programming because it facilitates decision-making and branching in code.

• **Boolean Data Type**: The boolean data type has only two possible values: `True` or `False`. These are used to control the flow of programs through conditionals and loops.
• **Logical Operations**: Common operations include AND, OR, and NOT. These operators combine boolean values to get a resultant boolean value. For instance, in a condition that checks if one number is a multiple of another, the result will always be a boolean - either indicating the numbers are multiples or they are not.

In the context of the exercise, the `multiple` function returns a boolean value by using a simple logical comparison. It checks `b % a == 0` and returns `True` if the condition is met, otherwise `False`, enabling the program to make further decisions based on this outcome.

Program Structure

A well-structured program generally consists of a clear flow and modular design. This involves organizing code into functions and using control structures to dictate how the program runs.

• **Functions**: Functions allow for reusability and modularity. They encapsulate a task, making it easier to maintain and reuse. In the given exercise, the `multiple` function is an excellent example of a reusable and testable piece of code that determines if a number is a multiple of another.
• **Control Flow**: Control structures such as loops and conditionals direct the execution flow of the program. For example, a loop can repeatedly perform an action until a condition is met, such as taking user input continuously until they choose to stop.
• **Input/Output**: A program generally interacts with users by taking inputs and providing outputs. This is essential for testing and running real-world scenarios.

The implementation in the exercise uses a while loop to handle multiple pairs of integers, demonstrating how control flow and function calls work together to achieve the desired result.