Is the number one even or odd?

As a domain expert in mathematics, I'm here to provide a comprehensive understanding of the concepts of even and odd numbers, which are fundamental to arithmetic and number theory. ### Introduction to Even and Odd Numbers Even and odd numbers are two distinct categories of integers that are defined based on their divisibility by the number 2. The distinction between them is straightforward and can be understood through the lens of their mathematical properties. #### Even Numbers Even numbers are integers that can be exactly divided by 2 without leaving a remainder. In mathematical terms, an even number can be expressed as \(2n\), where \(n\) is an integer. This means that for any integer \(n\), multiplying it by 2 will result in an even number. The sequence of even numbers starts with 2, 4, 6, 8, and so on, and extends infinitely in both directions. #### Odd Numbers Odd numbers, on the other hand, are integers that cannot be divided evenly by 2. They leave a remainder of 1 when divided by 2. Mathematically, an odd number can be represented as \(2n + 1\), where \(n\) is an integer. This indicates that if you take any integer \(n\) and add 1 to it, the result will be an odd number. The sequence of odd numbers begins with 1, 3, 5, 7, and continues indefinitely. ### The Number One: Even or Odd? Now, let's address the question at hand: Is the number one even or odd? To determine this, we can apply the definitions we've just discussed. - Even Numbers: An even number is divisible by 2, which means \(1 \div 2 = 0.5\), leaving a remainder of 0.5. - Odd Numbers: An odd number leaves a remainder of 1 when divided by 2, which means \(1 \div 2 = 0.5\) as well, but the key difference is that an odd number is of the form \(2n + 1\), which one is not. Given these definitions, it's clear that one does not fit the definition of an even number because it does not result from multiplying an integer by 2. However, it also does not fit the strict definition of an odd number because it is not of the form \(2n + 1\) for any integer \(n\). The number one is a unique case; it is neither even nor odd by the standard definitions. ### Conclusion In conclusion, while the definitions of even and odd numbers are clear, the number one stands in a category of its own. It is not divisible by 2 without a remainder, which disqualifies it from being even. At the same time, it does not conform to the formula for odd numbers. Therefore, one is considered a unique integer that does not fall into the traditional classification of even or odd. Now, let's proceed with the next steps as per your instructions.

An odd number is an integer of the form , where is an integer. The odd numbers are therefore ..., , , 1, 3, 5, 7, ... ... Integers which are not odd are called even. Odd numbers leave a remainder of 1 when divided by two, i.e., the congruence holds for odd .