\((\text { Multiples })\) Write an application that reads two integers, determines whether the first is a multiple of the second and prints the result. [Hint: Use the remainder operator.]

Understanding the remainder operator is essential when learning how to write code in Java. This operator is depicted as the percent sign (%), and it is used to find the remainder of the division of two numbers. Imagine we have the operation 7 % 3. The division of 7 by 3 gives us 2 with a remainder of 1. Here, the remainder operator will return 1 because it's the leftover after the integer division.

In our exercise, the remainder operator plays a crucial role. We're tasked with checking if one integer is a multiple of another. For instance, to determine whether 4 is a multiple of 2, we perform the operation 4 % 2, which would give us 0, meaning 2 divides 4 cleanly with no remainder, confirming that 4 is indeed a multiple of 2.

Do note, when the first number is smaller than the second, the result of the remainder operation is always the first number, since the division yields zero times with the given remainder. For example, 1 % 4 would give us 1, as 4 cannot divide into 1 any whole number of times.

Program logic is the series of steps or instructions within a program that are executed to produce the desired outcome. When we develop solutions for programming problems, creating a logic flow is akin to developing a recipe. Each step must be precise and clear to avoid any misinterpretation by the computer.

In the context of checking for multiples, the logic flow entails: prompting the user for two numbers, storing these numbers, applying the remainder operator, and then making a decision based on the result to inform the user.

• If the remainder is zero, the numbers are multiples.
• If the remainder is non-zero, they are not multiples.

Improving program logic often involves considering edge cases, such as what happens when a user enters a zero (since division by zero is undefined) or negative numbers, ensuring that the program handles such situations gracefully.

Integer division is a form of division in which the outcome is an integer rather than a decimal or fraction. In most programming languages, when you divide one integer by another, the result is automatically floored, i.e., rounded down to the nearest integer.

In our exercise problem, this concept is subtly present. We're more directly concerned with the remainder of such a division (using the remainder operator). However, understanding that 15 divided by 2 in integer division equals 7 with a remainder of 1 (15 % 2 equals 1) is fundamental. To say that an integer 'a' is a multiple of another integer 'b' means that 'a' divided by 'b' yields an integer with no remainder. So, integer division without a remainder indicates a multiple relationship between two numbers. Hence, in the exercise improvement advice, reaffirming the definitions and operations of integer division can aid in the deeper understanding of why and how the remainder operator determines multiplicity.