Let n be a positive whole number. Find three different proofs (as illustrated in Example 3 ) of

step 1 This problem, we want to show that n -cube minus n is divisible by 3 for all n in the positive i

Let n be a positive whole number. Find three different...

Key Concepts

Even and Odd Integers

This concept refers to the classification of integers based on divisibility by 2. An integer is even if it can be expressed as 2k for some integer k, and odd if it can be written in the form 2k + 1. This classification is fundamental in number theory and is frequently used in proofs involving divisibility and parity properties.

Direct Proof

A direct proof involves deducing the conclusion from the given hypothesis through a series of logical steps, using well-established definitions and previously proved theorems. It is often a straightforward method when the relationship between the hypothesis and the conclusion can be clearly established.

Proof by Contrapositive

In a proof by contrapositive, one proves that the contrapositive of the statement is true. That is, to prove that if A then B, one proves that if not B then not A. This method is sometimes more intuitive or simpler than a direct proof, and it is widely used when dealing with implications in mathematics.

Proof by Contradiction

Proof by contradiction assumes the opposite of what you want to prove and then demonstrates that this assumption leads to a logical inconsistency. The resulting contradiction confirms that the original statement must be true. This method is particularly powerful in number theory and other fields where direct approaches are less evident.

Properties of Squares in Number Theory

Understanding how squaring affects the parity and divisibility properties of integers is critical in number theory. For instance, if the square of an integer exhibits certain divisibility characteristics, one can often infer similar properties about the original number. This concept underpins many proofs concerning the relationship between a number and its square.

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