Kenneth Rosen Edition 7 Exercise 1.7 Question 3 (Page No. 91)

Question

Show that the square of an even number is an even number using a direct proof

Answer 1

To prove that the square of an even number is an even number, we need to demonstrate that for any even number n, n^2 is also an even number.

Let's assume that n is an even number. By definition, an even number can be expressed as 2k, where k is an integer.

So, we can write n as n = 2k.

Now, let's find the square of n:

n^2 = (2k)^2 = 4k^2.

Since k is an integer, k^2 is also an integer. Therefore, we can rewrite the equation as:

n^2 = 4k^2 = 2(2k^2).

From this expression, we can see that n^2 can be expressed as 2 multiplied by an integer (2k^2). By definition, this means that n^2 is an even number.