The reason that multiplying any integer by an even number always results in an even number lies in the properties of even numbers.

Explanation:

1. Definition of an Even Number: An even number is any integer that can be written in the form $2k$, where $k$ is an integer. This means that an even number is always divisible by 2.

2. Multiplying an Even Number by Any Integer: When you multiply an even number $2k$ by any integer $n$, the result is: $(2k) \times n = 2(kn)$ This shows that the result is $2 \times (kn)$, which is clearly a multiple of 2.

3. Conclusion: Since the result $2(kn)$ is divisible by 2, it is always an even number, regardless of the value of $n$.

Therefore, the key reason is that the factor of 2 in an even number is preserved in the product, making the result always even.

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Related Questions:

1. What is the definition of an odd number, and why does multiplying an odd number by an even number also result in an even number?
2. How does adding two even numbers together result in another even number?
3. What is the result when you multiply two odd numbers? Why?
4. Can an even number ever be the result of multiplying two odd numbers? Why or why not?
5. What happens when you divide an even number by 2? Will it always result in an integer?

Tip:

Always check for divisibility by 2 when determining if a number is even. If it’s divisible by 2, it is always even.