Why Multiplying an Even Number Always Results in an Even Number

What is the result of multiplying an even number by any number?

• A. Odd
• B. Even
• C. Positive
• D. Negative

Answer: (B)

When an even number is multiplied by any other number, the result is always even. An even number is defined as any integer that can be expressed in the form \( 2n \), where \( n \) is an integer. When you multiply this form by another integer \( m \), the product can be expressed as \( 2n \times m = 2(n \times m) \). Since \( n \) and \( m \) are both integers, their product \( (n \times m) \) is also an integer. Therefore, the final result is expressed as \( 2 \) multiplied by an integer, which confirms that the result is even. This principle holds true regardless of whether the other number is even, odd, positive, or negative. The key takeaway is that the even characteristic of the number is preserved in the product, thus leading to the conclusion that multiplying any even number by any number always yields an even result.

If you're preparing for the GMAT, you might’ve encountered some tricky questions on number properties. One fundamental concept that often comes up is what happens when you multiply even numbers. Ever wondered why the product of an even number with any other integer remains even? Let’s unravel this together—it's simpler than you might think!

First off, what exactly defines an even number? You could think of it as any integer that’s made up of ( 2n ), where ( n ) is an integer itself. For instance, take the number 4. We can express it as ( 2 \times 2 ); simple enough, right? So when you take this even number and multiply it by another integer, say 3, the math looks like this:

[

2n \times m = 2(n \times m)

]

What this equation shows is that you’ll always end up with a product that can be expressed in the form of ( 2 \times \text{integer} ). That means the result is, without a doubt, even. You know what? It's almost like magic—wherever you start with even, you always finish with even!

Now, you might ask: "What if the other number is odd or negative?" Great question! It doesn’t matter if you’re multiplying by an odd number, a negative number, or even another even number; the outcome will remain the same. Multiplying any even number protects this property, like a steadfast guardian—making sure that the result will always stay even. Isn't math fascinating when it behaves so predictably?

Let's illustrate this with a couple of practical examples to cement this idea. Picture multiplying 2 (even) by 5 (odd):

[

2 \times 5 = 10

]

Ten is an even number! How about we try it with a negative?

[

2 \times (-3) = -6

]

So there you have it—negative six is also even. It’s as if even numbers have a superpower that draws the result back into their familiar territory.

So, as you gear up for the GMAT, remember this essential principle: the even nature of a number doesn’t just fade into the background; it shapes the results of your calculations. This is not just a dry fact—it’s a straightforward concept that can help you answer questions on the exam with confidence.

Armed with this knowledge, you’ll tackle GMAT questions with a clearer understanding. And keep this in mind: math has a way of making the complex seem simple, almost cozy, once you get the hang of it. So why not embrace this comfort zone while you prepare? There's nothing like feeling empowered by your understanding of math, and who knows, you might even enjoy the process along the way!