Problem 26 - Elementary Algebra

Question

Determine the product of the first three positive odd integers.

Step-by-Step Solution

Answer

The product is 15.

Step 1: Identify the First Three Positive Odd Integers

The first three positive odd integers are 1, 3, and 5. Odd integers are numbers not divisible by 2.

Step 2: Set Up the Product

To find the product, set up the multiplication of these three integers: \( 1 \times 3 \times 5 \).

Step 3: Multiply the Numbers

First, multiply the first two numbers: \( 1 \times 3 = 3 \). Then, multiply the result by the third number: \( 3 \times 5 = 15 \).

Step 4: Confirm the Product

Verify the multiplication to ensure accuracy. Check that \( 1 \times 3 = 3 \) and \( 3 \times 5 = 15 \), confirming that the product of the three numbers is 15.

Key Concepts

Odd Integers

Odd integers are numbers that cannot be evenly divided by 2. If you try dividing an odd integer by 2, you will always have a remainder of 1. This unique characteristic makes them easily identifiable among other integers. Odd integers always appear alternately with even integers on the number line.

The sequence of odd integers begins at 1 and continues as 3, 5, 7, etc. Here are some qualities of odd integers:

• They are not multiples of 2.
• The difference between successive odd numbers is always 2.
• Odd integers can be negative as well as positive, such as -1, -3, -5, etc.

Understanding odd integers helps in tasks involving arithmetic operations and identifying patterns in number sequences.

Positive Integers

Positive integers are all the whole numbers greater than zero. They start from 1 and include numbers like 2, 3, 4, and so on infinitely. Positive integers do not include decimals, fractions, or negative numbers. They form the set of natural numbers often used for counting.

Here's what makes positive integers distinct:

• They are greater than zero.
• They do not include zero itself, unlike non-negative integers.
• They can be used to measure quantities and perform calculations in everyday math.

Understanding the nature of positive integers is crucial as they often come up in mathematical problems requiring counting and addition.

Step by Step Math Solution

A step by step math solution is essential for solving math problems systematically and accurately. This method ensures that no detail is overlooked, and each step paves the way to achieving the correct result. Let's explore why a step-by-step approach is beneficial:

• Clarifies Each Stage: Breaking down a problem helps in understanding each part distinctly.
• Avoids Mistakes: By taking it one step at a time, the likelihood of errors decreases.
• Enhances Learning: It facilitates deeper understanding and retention of mathematical concepts.

In our exercise, to determine the product of the first three positive odd integers, we followed a step-by-step process:

1. Identify the integers: 1, 3, and 5.
2. Set up multiplication: \(1 \times 3 \times 5\).
3. Compute in stages: \(1 \times 3 = 3\), then \(3 \times 5 = 15\).
4. Verify to ensure accuracy.

Such an approach ensures understanding and demonstrates math as a logical, sequenced process.