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To understand the equivalence of mathematical statements involving integers, it's essential to grasp the basic properties of integers. An integer is a whole number that can be positive, negative, or zero. Some important properties to remember include:

• **Addition/Subtraction**: The sum or difference of two integers is always an integer.
  
• **Multiplication/Division**: The product of two integers is always an integer. However, division between two integers might not result in an integer (e.g., 3 divided by 2).
• **Parity (Odd/Even)**: Every integer is either odd or even.
• **Sign**: An integer can be negative, zero, or positive.

Remember, zero is considered even and neutral in terms of positivity and negativity. Understanding these properties is fundamental when proving mathematical statements.

Odd and even numbers are classifications that depend on divisibility by 2.
An **even number** is any integer that can be written as **2m**, where **m** is an integer. This means even numbers include 0, 2, 4, 6, 8, etc. An **odd number**, on the other hand, is any integer that can be written as **2k + 1**, where **k** is an integer. Examples of odd numbers include 1, 3, 5, 7, etc.
Several properties are crucial when working with odd and even numbers:

• **Sum Properties**: The sum of two even numbers is even, the sum of two odd numbers is even, while the sum of an odd number and an even number is odd.
• **Product Properties**: The product of two even numbers is even, the product of two odd numbers is odd, whereas the product of an odd number and an even number is always even.

Understanding these properties can help break down complex problems into simpler elements.

In mathematics, proof techniques are methods used to establish the validity of statements. Some common proof techniques include:

• **Direct Proof**: This involves starting with known facts and using logical steps to arrive directly at the statement we want to prove.
• **Proof by Contradiction**: We assume the statement we want to prove is false and show that this assumption leads to a contradiction.
• **Inductive Proof**: Uses the principle of mathematical induction which includes a base case and an inductive step to prove an infinite series of statements.

The provided exercise uses a combination of direct proof and equivalence (a form of biconditional proof) to show that one statement implies another which in turn implies all others. For example, to prove statement **(i)** (**n² is odd**), the solution assumes **n** is odd and then squares it to show **n²** is indeed odd. This direct approach simplifies the problem by breaking it down into manageable parts.

When tackling mathematical proofs, always start by understanding each statement separately and then methodically work through each step. Every proof, no matter how complex, can be approached in this logical and structured manner.