Question

Write an application that calculates the product of the odd integers from 1 to 15.

Short answer

The product of the odd integers from 1 to 15 is 2027025.

Step-by-step solution

Identify Odd Integers Between 1 and 15

First, we need to list all the integers from 1 to 15. From these, we should identify which numbers are odd. Odd numbers are those that are not divisible by 2. The odd integers between 1 and 15 are 1, 3, 5, 7, 9, 11, 13, and 15.

Set Up the Product Calculation

To calculate the product of a series of numbers, multiply them together sequentially. We'll denote the product we are looking for as "P". Initially, set the product, P, to 1, as the identity element for multiplication.

Multiply the Odd Integers

Starting with P = 1, multiply it by each successive odd integer. Perform the calculations as follows: - P = 1 × 1 = 1 - P = 1 × 3 = 3 - P = 3 × 5 = 15 - P = 15 × 7 = 105 - P = 105 × 9 = 945 - P = 945 × 11 = 10395 - P = 10395 × 13 = 135135 - P = 135135 × 15 = 2027025

Finalize and Verify Calculation

Verify the last multiplication to ensure each step was performed correctly. Confirm that multiplying 135135 by 15 indeed gives 2027025. Thus, the product of all odd integers from 1 to 15 is 2027025.

Key concepts

Loop Structures

Loop structures are essential tools in programming that help automate repetitive tasks. In Java programming, loops allow you to execute a block of code multiple times. This is particularly useful when dealing with sequences or ranges of values, like in our exercise where we need to work with odd integers from 1 to 15.

Java supports several types of loop structures, but the most common ones are the `for` loop, the `while` loop, and the `do-while` loop. Each serves a specific scenario depending on the needs of the algorithm.

• The `for` loop is useful when the number of iterations is known beforehand. It's ideal for iterating over a series of numbers, as in our current scenario of listing odd numbers.
• The `while` loop is suitable when the condition might change within the loop and you may not know in advance how many times it will iterate.
• The `do-while` loop guarantees that the code block inside the loop will run at least once, even if the condition is false initially.

In the exercise, a `for` loop would be perfect because it can iterate through numbers 1 to 15, checking each to see if it is odd and then including it in our product calculation.

Integer Multiplication

Integer multiplication is a fundamental mathematical operation, particularly significant in computer programming. It involves multiplying two whole numbers to produce another whole number, known as the product. In our task, not only are we performing integer multiplication, but we are also doing so repeatedly over a series of odd numbers.

When multiplying integers in Java, it is important to handle the data types correctly to avoid overflow—when the product exceeds the maximum value that an `int` can hold. Java offers several numeric data types such as `int`, `long`, and `BigInteger`, each catering to different sizes of numbers.

• An `int` can represent a value up to 2,147,483,647, which is why it manages fine for our result of 2,027,025 in this exercise.
• If larger products were expected, a `long` or `BigInteger` would be more appropriate. `BigInteger`, for example, allows the handling of arbitrarily large numbers.

In our specific task, starting with a product `P` initialized to 1 is a strategic choice, as 1 is the identity element of multiplication, meaning it will not affect the final result when multiplied.

Algorithm Development

Algorithm development is about creating a step-by-step solution to a problem, structured in a way that a computer can execute. For this exercise, we need to find an efficient means to calculate the product of all odd integers from 1 to 15.

The core steps involved in developing an algorithm include:

• Identifying the problem: We needed to calculate the product of specific numbers (odd integers) within a given range.
• Planning a solution: Listing out steps that include iterating over the number range, identifying odd numbers, and multiplying them.
• Choosing the right tools: Selecting the appropriate loop (like `for` loop), which allowed seamless iteration through the numbers.
• Coding and testing: Implementing the code in a language like Java and ensuring it works through testing.

This approach applies not only to simple exercises like ours but also scales to more complex programming challenges. Systematically developing algorithms helps in optimizing time and resources, thereby leading to more efficient code. In programming, developing such systematic thinking is crucial to tackle both everyday and complex algorithmic challenges.