Problem 3 - Book of Proof

Question

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) If \(n \in \mathbb{Z}\) and \(n^{5}-n\) is even, then \(n\) is even.

Step-by-Step Solution

Answer

The statement is false. If \(n^{5} - n\) is even, it does not necessarily mean that \(n\) is an even integer.

Step 1: Understand the problem

Analyse the problem carefully to know what is given and what needs to be found. Here, the given part is that \(n\) belongs to integer and \(n^{5}-n\) is even. The target is to demonstrate whether or not \(n\) is indeed an even number.

Step 2: Pick an odd number and put it into the hypothesis

Remember that an odd number can be written in the form \(2k+1\), where \(k\) is an integer. Let's examine what happens when we choose an odd number and place it into the equation. If the statement is valid, we should get an odd result.

Step 3: Calculation

Now let's do the computation. If \(n = 2k + 1\), then \(n^{5} - n = (2k+1)^{5} - (2k+1)\). Use the binomial theorem to simplify this, and factor out a \(2k+1\). If you do the simplification, you'll find that it equals to \(2*(something)+0\), which is indeed even.

Step 4: Conclusion

So we put an odd number (which can be written in the form of \(2k + 1\), where \(k\) is an integer), into the equation, and ended up with an even number. This contradicts our initial assumption that if \(n^{5} - n\) is even, then \(n\) must be even. Therefore, the given statement is false.

Key Concepts

Integer Properties

Integers are the numbers consisting of zero, positive, and negative whole numbers. They are represented by the set \( \mathbb{Z} \), which includes numbers like -3, 0, 7, 42, and many others.
Understanding the properties of integers is crucial in solving mathematical proofs, particularly when evaluating statements involving even and odd numbers.
Some important properties of integers to consider include:

• Closure: The sum, difference, or product of any two integers is also an integer.
• Commutativity: For addition and multiplication, the order of integers does not matter, i.e., \( a + b = b + a \) and \( a \times b = b \times a \).
• Associativity: In addition and multiplication, the grouping of numbers does not change the result, i.e., \( (a + b) + c = a + (b + c) \).
• Distributive Property: Multiplication distributes over addition, such as \( a \times (b + c) = (a \times b) + (a \times c) \).

Recognizing these properties helps in simplifying complex expressions and proofs.

Binomial Theorem

The binomial theorem is a powerful tool in algebra that allows for the expansion of expressions raised to a power. It is expressed as:\[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \]This theorem is especially useful when dealing with expressions like \((2k + 1)^5\), which often appear in problems involving even and odd numbers.
By applying the binomial theorem, you can expand such expressions into a sum of terms. Each term in the expansion is composed of powers of 2 and an integer, helping to determine whether the final result is even or odd.
It's important to note:

• Factorials: The coefficients in the expansion \( \binom{n}{k} \) are calculated using factorials, \( \frac{n!}{k!(n-k)!} \).
• Simplification: After expansion, simplify the expression by grouping terms and factoring out common factors.
• Symmetry: The binomial coefficients \( \binom{n}{k} \) are symmetric, meaning \( \binom{n}{k} = \binom{n}{n-k} \).

Understanding how to effectively apply the binomial theorem helps in proving or disproving statements about integer properties.

Even and Odd Numbers

Even and odd numbers are fundamental concepts in mathematics.

• Even Numbers: An integer is even if it is divisible by 2, meaning it can be written in the form \( 2m \), where \( m \) is an integer.


• Odd Numbers: An integer is odd if it cannot be divided evenly by 2, meaning it can be written in the form \( 2k + 1 \), where \( k \) is also an integer.

Understanding these forms helps when working within mathematical proofs.
In the problem discussed, we analyze what happens when an odd number is used in a specific equation. We look to see if the result supports or contradicts the assumption that might imply \( n \) is even when \( n^5 - n \) is even.
Key points about even and odd numbers include:

• Sum and Difference: The sum or difference of two even numbers, or two odd numbers, is always even. The sum or difference of an even number and an odd number is always odd.
• Product: The product of an even number with any integer is always even. The product of two odd numbers is odd.

Recognizing these patterns is essential for logically constructing mathematical arguments and proofs.