Question

a whole number a is even if and only if there is a whole number k such that a = 2k. use this representation of an even number to prove that the square of an even number is divisible by 4.

selected options: a, b
a. the starting number is even.
b. every even number can be written in the form 2k where k is any whole number.
c. the square of the starting number is even.
d. the square of the starting number is divisible by 4.

what process can be used to prove that the square of the number is divisible by 4?
a. write an expression for any odd number, square it, and show it is not divisible by 4.
b. write an expression for any natural number, square it, and show it is not divisible by 4.
c. write an expression for any even number, square it, and show the result is divisible by 4.
d. write an expression for any natural number, square it, and show it is divisible by 4.

let variables represent any whole number.
then the expression 2k represents any and all even numbers.
the square of this expression is .
(simplify your answer.)

Explanation:

Identify the starting assumptions

Using the Even Number Definition knowledge point

$$ a = 2k \quad (k \in \mathbb{W}) $$

Determine the proof process

Using the Direct Proof and Algebraic Proofs knowledge points
To prove that the square of any even number is divisible by 4, we must:

1. Write a general algebraic expression for any even number, which is \(2k\).
2. Square this expression to obtain \((2k)^2\).
3. Show that the resulting expression is algebraically divisible by 4.

Thus, the correct process is described by option C: "Write an expression for any even number, square it, and show the result is divisible by 4."

Square the algebraic expression

Using the Algebraic Proofs knowledge point

$$ (2k)^2 = 2^2 \cdot k^2 = 4k^2 $$

Answer:

Question 1

• (A) The starting number is even. (Correct answer)
• (B) Every even number can be written in the form 2k where k is any whole number. (Correct answer)
• (C) The square of the starting number is even.
• (D) The square of the starting number is divisible by 4.

Question 2

• (A) Write an expression for any odd number, square it, and show it is not divisible by 4.
• (B) Write an expression for any natural number, square it, and show it is not divisible by 4.
• (C) Write an expression for any even number, square it, and show the result is divisible by 4. (Correct answer)
• (D) Write an expression for any natural number, square it, and show it is divisible by 4.

Question 3

The square of this expression is <blank>\(4k^2\)</blank>