Problem 38 - Fundamentals of Mathematics

Question

Suppose someone wants to find three consecutive odd integers that add to \(120 .\) Why will that person not be able to do it?

Step-by-Step Solution

Answer

Three consecutive odd integers cannot add to 120, as this results in even integers.

Step 1: Define the Consecutive Odd Integers

Let's define three consecutive odd integers. We can denote the first odd integer as \(x\). The next consecutive odd integer would be \(x + 2\), and the one after that would be \(x + 4\).

Step 2: Set Up the Equation

The problem states that the sum of these three integers is 120. We can write an equation for the sum:\[x + (x + 2) + (x + 4) = 120\] Simplifying this, we get:\[ 3x + 6 = 120\]

Step 3: Solve the Equation for x

To solve for \(x\), first subtract 6 from both sides:\[3x + 6 - 6 = 120 - 6\]which simplifies to:\[3x = 114\]Next, divide both sides by 3:\[x = 38\]

Step 4: Check Consecutive Odd Integer Property

Now, substitute \(x = 38\) to find the integers. The integers would be 38, 40, and 42. These numbers are not odd since they are all even. Therefore, \(x + 2\) and \(x + 4\) are also even integers.

Key Concepts

Integer Properties

Integers are an essential part of mathematics, comprising whole numbers that can be positive, negative, or zero. They have unique properties that help us solve problems involving them. When dealing with consecutive integers, remembering that each integer moves in consistent steps from the previous one is important.
For odd integers, this step is always two. An odd integer is any integer that cannot be exactly divided by two. If you start with an odd integer, the next one in line would be two units away. This is because adding two to an odd integer gives you another odd number.
When working with integers in equations, especially consecutive ones, it's helpful to use variable expressions to define them. A common approach is to define the first integer as a variable, say, "\(x\)," and express others in terms of this variable using addition, such as \(x + 2\) and \(x + 4\). This helps capture their relationships effectively in mathematical terms.

Solving Equations

Understanding how to solve equations is a fundamental skill in algebra. This involves finding the value of unknown variables that satisfy the given equation. In the context of our problem, we look to find an integer \(x\) that helps identify subsequent integers.
Step by step, solving equations generally involves:

• Combining like terms, such as in the equation \(x + (x + 2) + (x + 4) = 120\), which simplifies to \(3x + 6 = 120\).
• Isolating the variable, achieved by performing operations like subtraction or division on both sides of the equation. Here, subtract 6 from both sides to get \(3x = 114\).
• Solving for the variable by performing the final operation, such as division. Divide by 3 in this case, yielding \(x = 38\).

Mastering these steps will allow you to find solutions efficiently and accurately.

Odd Numbers

Odd numbers are integers that are not divisible by two. Throughout mathematics, they present a distinct character, especially when approaching problems involving patterns or sequences. Examples of odd numbers include 1, 3, 5, 7, and so forth.
When tasks require finding odd integers, knowing their base property - non-divisibility by two - is key. Two added to any odd number results in another odd integer. This reflects their predictable pattern.
In our original exercise, even though the integer \(x\) was found to be 38, the subsequent integers (38, 40, 42) turned out to be even, contradicting the requirement for odd integers. Hence, recognizing that the sum specified cannot be met by odd integers reinforces the importance of distinguishing even and odd properties upfront when analyzing and validating potential solutions.