Verified Answer :
- An integer \(k\) belongs to the set \(\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}\).
- An even number is defined as any integer \(n\) that can be written in the form \(n = 2m\) for some integer \(m\).
- The product \(2k\) directly fits this definition by substituting \(m = k\), where \(k\) is an integer.
- Therefore, the product of 2 and any integer is always an even number, which can be stated as \(\boxed{\text{even}}\).
Explanation
- Analysis and Definitions
The problem asks to explain why the product of 2 times any integer results in an even number. An integer, denoted as \(k\), is any element in the set of integers \(\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}\). We are looking to show that for any integer \(k\), the product \(n = 2k\) is always an even number.
- Definition of an Even Number
The mathematical definition of an even number is central to this problem. An integer \(n\) is classified as even if and only if there exists some integer \(m\) such that \(n = 2m\). This means that any number that can be expressed as 2 multiplied by another integer satisfies the condition for being even.
- Formal Proof
Consider the product of 2 and an arbitrary integer \(k\). Let this product be \(n\), so \(n = 2 \times k = 2k\). Since \(k\) is an integer by definition, and the definition of an even number states that any integer that can be written in the form \(2m\) for some integer \(m\) is even, we can identify \(m\) as \(k\). Since \(k\) is an integer, the expression \(2k\) directly matches the form \(2m\). Therefore, \(n = 2k\) is an even number.
- Conclusion and Final Answer
In conclusion, for any integer \(k\), the product \(2k\) satisfies the definition of an even number. Thus, the product of 2 and any integer is always an even number. The final answer is that the product is even.
Examples
Understanding the property of even numbers is crucial in various fields, such as computer science where data is often represented in binary. For instance, in bitwise operations, multiplying a binary number by 2 is equivalent to a bit-shift operation to the left (e.g., \(3 \times 2 = 6\), or in binary 11 becomes 110), which ensures the last bit is always 0, confirming the result is even. This concept is a foundation for parity checks in error detection systems to ensure data integrity during transmission.