Question

(Calculating the Product of Odd Integers) Write an application that calculates the product of the odd integers from 1 to 15.

Short answer

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

Step-by-step solution

Understanding the Task

The task is to calculate the product of all odd integers from 1 to 15. This entails identifying all the odd numbers between 1 and 15 and then multiplying them together.

Identify Odd Integers from 1 to 15

List out all the odd integers in the given range. These are 1, 3, 5, 7, 9, 11, 13, and 15.

Calculate the Product

Multiply all the identified odd integers together. The operation will be as follows: \( 1 \times 3 \times 5 \times 7 \times 9 \times 11 \times 13 \times 15 \).

Compute the Multiplication

Performing the multiplication, we get the product: \( 1 \times 3 = 3 \), \( 3 \times 5 = 15 \), \( 15 \times 7 = 105 \), \( 105 \times 9 = 945 \), \( 945 \times 11 = 10395 \), \( 10395 \times 13 = 135135 \), and finally \( 135135 \times 15 = 2027025 \). Therefore the product of all odd integers from 1 to 15 is 2027025.

Key concepts

Odd Integers Multiplication

Understanding multiplication of odd integers is crucial for tackling various mathematical problems in programming. An odd integer is any number that cannot be exactly divided by 2, and it has a remainder of 1 when divided by 2. Examples of odd integers include 1, 3, 5, 7, and so on.

When we multiply odd integers, we are essentially performing a series of arithmetic operations where the result will also be an odd integer. In Java, or indeed in any programming language, we can systematically perform multiplication operations to calculate the product. In the context of our problem, calculating the product of odd integers from 1 to 15 involves multiplying all the odd numbers within that range. When done sequentially, as illustrated in the step-by-step solution, you'll notice the product grows exponentially, which is a common characteristic of the multiplication of integers.

Java for Loop

The Java for loop is a control flow statement that allows code to be executed repeatedly based on a boolean condition. It's an efficient way to iterate through a range of numbers, a collection, or an array.

For example, to iterate through odd numbers from 1 to 15, a 'for' loop is ideal. The loop starts with an initial value, continues to execute the statement block as long as the condition is true, and incrementally updates the value.

Sample Code Structure for Odd Numbers

for (int i = 1; i <= 15; i += 2) {  // body of loop}

The loop above uses the initialization block to start with the first odd number, the condition to ensure the loop doesn't exceed 15, and the iteration statement to jump to the next odd number (by adding 2 every time). This avoids any need to check if the number is odd, as the loop inherently moves through the odd numbers only.

Java Arithmetic Operations

Arithmetic operations constitute the backbone of problem solving in Java, including operations such as addition, subtraction, multiplication, and division. In our example, the relevant operation is multiplication. Java follows the order of operations from mathematics, so keeping track of the operations and the order in which they are performed is essential for correct results.

When coding in Java, these operations use familiar symbols: '+', '-', '*', '/'. For multiplying odd integers in a loop, the multiplication operator '*' is used in conjunction with an assignment operator to update the product, as you can see in the iterative steps of the solution. An important aspect to remember is managing the data types, such as using 'int' or 'long' to store the result, ensuring that the variable can hold the final product without causing an overflow error.