Problem 8 - Discrete Mathematics with Applications

Question

Prove each directly. The product of any two even integers is even.

Step-by-Step Solution

Answer

To prove that the product of any two even integers is even, let \(a = 2m\) and \(b = 2n\) be two even integers, where \(m\) and \(n\) are any integers. Their product \(c = a \cdot b = (2m)(2n) = 4mn = 2(2mn)\). Since \(c\) can be represented as 2 times another integer \((2mn)\), the product of any two even integers is also even.

Step 1: Define even integers

An even integer is defined as an integer that can be represented as 2 times another integer (this other integer does not have to be even or odd). Mathematically, we can represent this as: \( a = 2m \) and \( b = 2n \) where \(a\) and \(b\) are even integers and \(m\) and \(n\) are any integers.

Step 2: Calculate the product

Now, we'll find the product of these two even integers \(a\) and \(b\). \(c = a \cdot b\) Since \(a = 2m\) and \(b = 2n\), we can substitute these expressions into the equation above: \(c = (2m)(2n)\)

Step 3: Simplify the product

Now, let's simplify the product of these two factors: \(c = 4mn\)

Step 4: Show that the product is even

Notice that we can rewrite the expression for the product \(c\) as follows: \(c = 2(2mn)\) Since the product \(c\) can be represented as 2 times another integer \((2mn)\), this confirms that the product of any two even integers is also even. Therefore, the statement is proven.

Key Concepts

Product of Integers

When multiplying integers, the outcome is known as the product of those integers.

For even integers, this often involves probing deeper to understand what it means for a number to be even and how it influences multiplication. Since an even integer can be written as two times another integer, it follows that when you multiply two such numbers, you are effectively multiplying groups of twos.

• For instance, given two even integers represented as \(a = 2m\) and \(b = 2n\), their product \(c = a \cdot b\) would be \((2m) \cdot (2n)\).
• This product simplifies to \(4mn\), which is clearly a multiple of two, hence is even.

Understanding that the result \(c = 4mn\) keeps the property of evenness intact showcases why the multiplication of two evens always results in another even integer.

Proof by Direct Method

A proof by direct method is a fundamental approach in mathematics where we demonstrate a statement's truth step-by-step.

This method involves taking known facts and applying them to reach the desired conclusion without assuming the opposite.

• We start by defining what an even integer is: any number which can be expressed as \(2\times \text{an integer}\).
• Then, using these definitions, one calculates and simplifies to see if the required property holds.

In the example of proving that the product of two even integers is even, the proof starts with expressing each even integer as \(2m\) and \(2n\), respectively.
The direct approach continues by calculating their product and verifying that it can still be described in terms of two times an integer, confirming that it is indeed even.

Properties of Even Numbers

Even numbers have fascinating properties that aid in mathematical proofs and problem-solving. Recognizing these properties is crucial in discerning how even numbers behave under various operations.

• One key property is that any integer multiplied by two results in an even number, as seen in expressions like \(2m\).
• Another is that even numbers can be added or multiplied, and they will yield an even number, which is particularly essential for direct proofs.

Within multiplication, like when proving a product of even integers, these properties help simplify calculations and affirm outcomes. By understanding these fundamental aspects, we can more easily predict and affirm behaviors in arithmetic operations.

Integer Multiplication

Integer multiplication is a core operation in mathematics involving combining two or more integers to yield a single product. This process follows specific rules and properties, especially when dealing with subsets of integers such as even numbers.

• When dealing with even integers, each number can be expressed in a form \(2m\), maintaining its characteristic quality of being divisible by two.
• Upon multiplication, the result, for instance, \( (2m)(2n) = 4mn \), remains tied to its origins, as the structure inherently preserves divisibility by two, indicating evenness.

Understanding how integer multiplication operates, specifically with even numbers, is vital for handling more complex mathematical scenarios where the behaviour of even and odd numbers dictates the outcome.