Problem 1 - A Gentle Introduction to the Art of Mathematics

Question

Prove that if the cube of an integer is odd, then that integer is odd.

Step-by-Step Solution

Answer

If the cube of an integer is odd, the integer itself must be odd.

Step 1 - Define an Odd and Even Integer

An integer is considered odd if it can be written in the form of \(2k + 1\), where \(k\) is an integer. An integer is considered even if it can be written in the form of \(2k\).

Step 2 - Cube of an Odd Integer

First, let's consider the cube of an odd integer. If an integer \( n \) is odd, it can be written as \( n = 2k + 1 \). Calculate its cube: \[ (2k + 1)^3 = 2k + 1 \times 2k + 1 \times 2k + 1 \]

Step 3 - Simplify the Expression

Expand the cube expression and simplify: \[ (2k + 1)^3 = 2k + 1 \times 4k^2 + 4k + 1 \] \[ = 8k^3 + 12k^2 + 6k + 1 \] As we can see, the result is \(8k^3 + 12k^2 + 6k + 1\), which can be written as \(2m+1\), where \(m\) is some integer.

Step 4 - Conclusion for Odd Integer

Since the cube of an odd integer \( (2k+1)^3 \) results in another odd integer \(2m+1\), we see the expression always results in an odd number.

Step 5 - Consideration of Even Integers

Let's consider the cube of an even integer. If an integer \( n \) is even, it can be written as \( n = 2k \). Calculate its cube: \[ (2k)^3 = 8k^3 \]. Since 8k^3 is clearly an even number \(2(4k^3)\), the cube of any even number is even.

Step 6 - Contradiction if Cube is Odd

Given that an even number's cube is always even, if an integer \( n \) has an odd cube, it implies that \( n \) itself must be odd (because if it were even, its cube would be even, giving us a contradiction). Thus, if \( n^3 \) is odd, then \( n \) must be odd.

Key Concepts

integer properties

Integers are whole numbers that can be positive, negative, or zero. There are two key types of integers: odd and even. An odd integer can be written in the form \(2k + 1\), where \(k\) is any integer. An even integer can be written as \(2k\). This means:

• Odd: 1, 3, 5, 7
• Even: 2, 4, 6, 8

Properties of integers are useful in proving conditions or deriving new expressions.

proof techniques

When proving mathematical statements, structured techniques help. One common technique is **direct proof**. Here, we use logical steps to show that something is true:

• Start by assuming the conditions given in the problem
• Apply definitions and known properties
• Draw conclusions step-by-step

For example, to prove the cube of an odd integer is always odd, we:
1. Assume an integer is odd: \(2k + 1\)
2. Cube it and expand: \((2k + 1)^3 = 8k^3 + 12k^2 + 6k + 1\)
3. Recognize the result is another odd number because it's in the form \(2m + 1\).
Such systematic proofs help to verify statements logically and clearly.

mathematical reasoning

Mathematical reasoning is the process of using logical thinking to solve problems. It's essential for proofs and problem-solving. Here's how it works:

• **Understanding the problem**: Clearly identify what needs to be proven.
• **Breaking down the problem**: Break it into smaller parts based on definitions and properties.
• **Using logical steps**: Apply known rules and logical steps to derive the solution.

For instance, in proving that an integer with an odd cube must be odd, reasoning involves:
1. Recognizing even integers have even cubes (\(2k)^3 = 8k^3\).
2. Establishing that since the cube is odd, the integer itself must be odd (creating a contradiction if assumed otherwise).
This method of reasoning ensures that the conclusions drawn are solid and based on established principles.