Problem 89 - Elementary Algebra Within Reach

Question

Determine whether the statement is true or false. Justify your answer. The sum of an odd integer and an even integer is even.

Step-by-Step Solution

Answer

The statement is false. The sum of an odd integer and an even integer is odd.

Step 1: Understanding Odd and Even Integers

An even number is one which can be written in the form 2n, where n is an integer. An odd number can be written in the form 2m+1, where m is also an integer.

Step 2: Adding an odd and an even number

To figure out if the sum of an odd and an even number is even, take an even number represented by \(2n\) and add it to an odd number represented by \(2m+1\).

Step 3: Simplification

When we add the two numbers \(2n + 2m +1\), it simplifies to \(2(n+m) + 1\). Here, \((n+m)\) is also an integer, say z. Hence, the sum of an even number and an odd number can be represented as \(2z + 1\). This is the form of an odd number. Therefore, the sum of an even number and odd number is odd.

Step 4: Conclusion

According to the conclusion drawn from the given examples, the statement that the sum of an odd integer and an even integer is even is proven to be false.

Key Concepts

Algebra and Integer Classification

In algebra, learning how to classify and work with integers—whole numbers including positive, negative, and zero—is foundational. An essential aspect of this includes understanding the distinction between even and odd integers. An even integer is one that is perfectly divisible by 2 without leaving a remainder. It can be described by the algebraic expression \(2n\), where \(n\) is any integer. For example, numbers such as 0, 2, -4, and 10 are even since they can be written as \(2 \times 0\), \(2 \times 1\), \(2 \times (-2)\), and \(2 \times 5\) respectively.

An odd integer, conversely, will have a remainder of 1 when divided by 2. This can be represented by the formula \(2m+1\), with \(m\) being an integer. Examples include -1, 3, 7, and 15, which correspond to \(2 \times (-1) + 1\), \(2 \times 1 + 1\), \(2 \times 3 + 1\), and \(2 \times 7 + 1\) respectively. The ability to identify and express integers as even or odd is a stepping stone toward more complex arithmetic operations and algebraic concepts.

Properties of Integers in Arithmetic

Integer properties are fundamental to understanding arithmetic operations. These properties allow us to predict and determine the outcomes of operations such as addition, subtraction, multiplication, and division. A primary consideration is the parity of integers, which refers to the classification of an integer as either even or odd.

Let's delve further into addition. When two even integers are added together, the sum is also even. For instance, adding \(2n\) to \(2m\) yields \(2n + 2m\), which is \(2(n + m)\) - clearly an even number. Similarly, when two odd integers are added, the result will also be even, since \(2m+1\) plus \(2n+1\) equals \(2(m + n + 1)\), another even number. However, the addition of an even integer and an odd integer, such as \(2n\) and \(2m+1\), results in an expression like \(2n + 2m + 1\), which simplifies to \(2(n + m) + 1\)—an odd number. Understanding these outcomes is essential in both simple and complex mathematical problem solving.

Arithmetic Operations with Integers

Arithmetic operations with integers, such as addition, subtraction, multiplication, and division, obey certain rules and behaviors depending on the nature of the integers involved. The previous discussion on integer properties pointed out how the sum of an even and an odd integer results in an odd integer. This observation can be extended to other operations as well.

For multiplication, an even number multiplied by any integer will always yield an even result \(2n \times m = 2(nm)\), regardless if \(m\) is odd or even. However, an odd number multiplied by an odd number will always result in an odd product \( (2n+1) \times (2m+1) = 2(2nm+ n + m) + 1 \). These consistent behaviors are due to the underlying integer properties and the structure of arithmetic operations. In division, things can get a bit more complex as division by zero is undefined and the quotient of two integers may not always be an integer. It is important to note that for an integer to divide another and result in an integer, it must be a factor of the number being divided. For example, \(6 \div 3 = 2\) is valid, but \(5 \div 2\) yields a non-integer result, highlighting the importance of understanding and applying the correct arithmetic operations to maintain the set of integers.