In Exercises 15-17 , decide whether the conjecture is true...

In Exercises $15-17$ , decide whether the conjecture is true or false. Try to give a convincing proof of the conjectures that are true. For false conjectures, give a counterexample.
The value of $n^{2}-n$ is always an even number.

Key Concepts

Mathematical Proof Techniques

Mathematical proofs are logical arguments that demonstrate the truth or falsity of a conjecture. Common techniques include direct proof, proof by contradiction, and proof by counterexample. In problems involving conjectures, these strategies help validate or refute the asserted statements.

Properties of Consecutive Integers

Consecutive integers are sequential numbers where each number is one more than the previous one. An important property of consecutive integers is that one of them is always even, which implies that their product is even. This reasoning is often used in proofs involving expressions that involve consecutive factors.

Conjectures

A conjecture is an unproven statement that is generally believed to be true based on heuristic arguments or empirical evidence. The process of confirming a conjecture involves providing a rigorous proof, whereas disproving it requires finding a counterexample that shows the statement does not hold in all cases.

Even and Odd Numbers

Even numbers are integers that are divisible by 2 without leaving a remainder, while odd numbers are not. Understanding the properties of even and odd numbers is crucial in identifying whether expressions yield even or odd results, especially when using algebraic manipulations.

Factorization of Algebraic Expressions

Factorization involves rewriting an algebraic expression as a product of simpler expressions. In many cases, factorization can reveal inherent properties of the expression, such as its divisibility. Recognizing that an expression can be factored into a product of integers may help in showing that it is even or has another property.

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