Problem 11 - Discrete Mathematics with Applications

Question

Prove each directly. The product of any even integer and any odd integer is even.

Step-by-Step Solution

Answer

To prove the product of any even integer and any odd integer is even, let the even integer be represented as \(2a\) and the odd integer be represented as \(2b + 1\), where a and b are integers. Multiply them together, \((2a)(2b + 1)\), and apply the distributive property, resulting in \(4ab + 2a\). Factor out the common factor of 2 from the expression: \(2(2ab + a)\). Since \(2ab + a\) is an integer, the entire expression is of the form \(2c\), where \(c\) is an integer (\(c = 2ab + a\)). Hence, the product of any even integer and any odd integer is always even.

Step 1: Define even and odd integers in general form

An even integer can be represented as 2a, where a is any integer. An odd integer can be represented as 2b + 1, where b is any integer.

Step 2: Multiply the even and odd integers

Now, let's multiply the even and the odd integers together: (2a)(2b + 1)

Step 3: Distributive property

Apply the distributive property to multiply the numbers: = 2a * 2b + 2a * 1

Step 4: Simplify the expression

Now, combine the terms: = 4ab + 2a

Step 5: Factoring out the common factor of 2

We can notice that there's a common factor of 2 in both terms, so we can factor it out: = 2(2ab + a)

Step 6: Conclude the proof

Since 2ab + a is an integer (as it is the sum of the product and an integer), the entire expression is of the form 2c, where c is an integer (c = 2ab + a). Therefore, the product of any even integer and any odd integer is always even.

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to multiply a number by a sum. This can be particularly helpful when dealing with expressions like \((2a)(2b + 1)\).

• The property states that multiplying a number by a sum is the same as multiplying each addend by the number and then summing the results.
• In mathematical terms, it can be expressed as: \(a(b + c) = ab + ac\).

In the context of the problem, the distributive property enables the transformation: \(2a(2b + 1)\) to \((2a \cdot 2b) + (2a \cdot 1)\). This property is essential for expanding and simplifying the expression, making it easier to work with integers and prove the original statement.

Integer Multiplication

Integer multiplication is a straightforward arithmetic operation. When multiplying integers, you combine each term as in the problem, where an even integer \(2a\) is multiplied by an odd integer \(2b + 1\).

• This operation takes the form \(2a \times 2b + 2a \times 1\).
• Here, the process involves straightforward multiplication of coefficients and constants.

This step sets the stage for applying further algebraic rules, like the distributive property, to simplify the expression.It's crucial to ensure each multiplication step is handled accurately for a correct outcome in more complex problems.

Factorization

Factorization is a process where you express a number or an expression as a product of its factors. This becomes a vital step in simplifying algebraic expressions, especially when proving statements about even and odd integers.

• In the solution, after applying the distributive property, we have the expression \(4ab + 2a\).
• Notice both terms share a common factor of 2, allowing us to factor it out: \(2(2ab + a)\).

By recognizing and factoring out common multiples, we simplify expressions and confirm or "see" that the result \(2(2ab + a)\) is indeed an even number,providing a clear path to the solution.

Proof Techniques

Proof techniques are methods used to establish the truth of a mathematical statement. In this exercise, direct proof is used. Let's break it down:

• A direct proof begins by assuming the initial premise (in this case, that we have an even and an odd integer) and logically derives the conclusion.
• Each step follows logically from the last, without assuming the conclusion.

In our problem, we start with expressions representing even and odd integers, apply algebraic rules like the distributive property, and come to the conclusion that the result, \(2(2ab + a)\), is even. Direct proofs are valued for their straightforward and logical sequence of steps, making them essential for learning fundamental problem-solving skills in mathematics.