Question

Use the method of direct proof to prove the following statements. If two integers have opposite parity, then their product is even.

Short answer

From the work carried out, it is evident that if two integers have opposite parity, their product has always resulted in an even number. This satisfies the original claim fully, and therefore it has been proven to be true. The short answer is - 'Yes, the product of two integers with opposite parity is always even.'

Step-by-step solution

Define Terms

First, understand that parity refers to whether an integer is even or odd. An even number can be represented as \(2k\) where \(k\) is any integer, while an odd number can be represented as \(2m + 1\) or \(2m - 1\) where \(m\) is an integer. In this problem, we have two integers with opposite parity, meaning one integer can be represented as \(2k\) while the other can be represented as \(2m + 1\) or \(2m - 1\).

Formulate Expression

Now, let's formulate the expression for the product of the two integers. Suppose the integer represented as \(2k\) is 'a' and the integer represented as \(2m + 1\) or \(2m - 1\) is 'b'. Their product \(ab\) can therefore be represented as \((2k)(2m + 1)\) or \((2k)(2m - 1)\).

Simplify Expression

Simplifying the expressions \((2k)(2m + 1)\) or \((2k)(2m - 1)\), we get \(4km + 2k\) or \(4km - 2k\) respectively. We can clearly write these expressions as \(2(2km + k)\) or \(2(2km - k)\).

Verify Evenness

Looking at the simplified expressions, we can clearly see that in both cases, the product is a multiple of 2, which means that the product is even. Hence, the product of two integers with opposite parity is indeed always even.

Key concepts

Proof in Mathematics

Proof is the cornerstone of mathematics, serving as the definitive tool to establish the truth of mathematical statements unequivocally. It's a logical argument that begins with known facts, applies mathematical principles, and leads to the conclusion we're trying to demonstrate.

In the context of our exercise, the method of direct proof was used, which is among the most straightforward proof techniques. To directly prove a statement, you assume the statement's premises are true and then use logical deduction to show the conclusion must follow.

For instance, to prove that the product of two integers with opposite parity is even, one begins by assuming that the integers indeed have opposite parity—one even, one odd—as defined. Then, by using algebraic manipulation and inherent properties of even and odd numbers, one arrives at a conclusion that matches the definition of an even number.

Parity of Integers

Parity refers to the classification of an integer as either even or odd. This concept is fundamental in various mathematical proofs and computations. An integer's parity provides insight into its divisibility by two and its behavior in operations such as addition, subtraction, and multiplication.

Understanding parity is not only important in theoretical mathematics but also in practical problems, such as algorithm design in computer science, where operations might have different efficiencies depending on the parity of the numbers involved.

Even and Odd Integers

Let's dive deeper into the world of even and odd integers. An even integer is any integer that can be divided by two without leaving a remainder and is typically represented by the formula: \(2k\), where \(k\) is any integer. Conversely, an odd integer cannot be divided by two without a remainder and can be represented as \(2m + 1\) or \(2m - 1\), where \(m\) is an integer.

These representations are not just abstract concepts; they are powerful tools in demonstrating mathematical properties like in the exercise at hand. They allow us to manipulate and combine numbers in proofs to showcase inherent structures, such as the existence of common factors—like the factor of two in the product of an even and an odd integer.