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Odd integers are numbers that are not divisible by 2. In other words, when you divide an odd number by 2, you get a remainder of 1. Examples of odd integers include 1, 3, 5, 7, and 9. Mathematically, we can express any odd integer as \(2k + 1\), where \(k\) is an integer.

This expression means that if you start from an even number \(2k\) and add 1 to it, you will get an odd number. For instance, if \(k = 2\), then \(2k + 1 = 2 \cdot 2 + 1 = 5\), which is an odd number.

When dealing with problems involving odd integers, such as adding them together or multiplying them, recognizing this basic expression \(2k + 1\) is crucial. It simplifies our calculations and helps prove various properties of odd numbers.

Even numbers are integers that are divisible by 2 without any remainder. Examples of even numbers include 2, 4, 6, 8, and 10.

Mathematically, any even number can be represented as \(2n\), where \(n\) is also an integer. This representation makes it easy to identify and work with even numbers. For instance, if \(n = 3\), then \(2n = 2 \cdot 3 = 6\), which is an even number.

One important property of even numbers is that when you add or subtract even numbers, the result is always even. For example, adding two even numbers, like \(6 + 4\), gives \(10\). Similarly, multiplying an even number by any integer will always result in an even number.

Integer factorization involves expressing an integer as a product of other integers, which are called its factors. For instance, the factors of 6 are 1, 2, 3, and 6 because \(6 = 2 \cdot 3\).

In the context of our exercise, factorization played a critical role in simplifying the sum of two odd integers. After adding the two expressions for odd integers \((2k + 1) + (2m + 1)\), the result was simplified to \(2(k + m + 1)\).

This factorization helped us conclude that the sum is even, as \(2(k + m + 1)\) is divisible by 2. Whenever we highlight common factors within algebraic expressions, factorization becomes a valuable tool. It simplifies our work and helps to understand the underlying properties of numbers and their operations.

It's good practice to familiarize yourself with recognizing and extracting common factors, as it proves helpful in a variety of mathematical problems.