Even Number Property and Its Implications

If x is an even number, then $x^2$ is an even number.

This problem is a fundamental exercise in understanding properties of even numbers and their behavior under certain mathematical operations, specifically squaring. Even numbers are integers that are divisible by two without any remainder. The intuitive concept here is that when an even number is squared, it results in a product that is also an even number. This property is derived from the fact that multiplying any integer by an even number results in an even number, which is a concept rooted deeply in number theory.

To solve this problem, one commonly begins with the definition of an even number. An even number, x, can be expressed in the form of $2k$ where k is an integer. When we square x, or $2k$, the expression becomes $(2k)^2$. Expanding this gives us $4k^2$, which can be rewritten as $2 imes (2k^2)$, clearly showing that the result is divisible by two, making it even.

Understanding problems like this not only reinforces foundational number theory concepts but also builds problem-solving skills in identifying patterns and applying definitions. Such exercises prepare students for more complex scenarios where these basic principles form the building blocks of more advanced mathematical reasoning.