Question

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

first, identify all the information that is known or given. check all that apply.
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 dropdown represents any and all dropdown numbers.

Explanation:

Identify the given information

Using the Even Number Definition knowledge point

• The problem statement begins: "A whole number \(a\) is even if and only if there is a whole number \(k\) such that \(a = 2k\)."
• It then asks to prove that the square of this number is divisible by 4.
• Therefore, the starting number \(a\) is known to be even, and any even number can be represented as \(2k\) for some whole number \(k\).
• Thus, the correct options for the first part are A and B.

Determine the proof process

Using the Algebraic Proofs knowledge point

• To prove that the square of an even number is divisible by 4, we must start with a general representation of an even number.
• We then square this expression and algebraically show that the resulting expression has a factor of 4.
• Therefore, the correct process is to write an expression for any even number, square it, and show the result is divisible by 4.
• This corresponds to option C.

Formulate the algebraic expression

Using the Even Number Definition knowledge point

• Let \(k\) represent any whole number.
• An even number is represented by the expression \(2k\).
• Therefore, the expression \(2k\) represents any and all even numbers.

Answer:

Question 1

First, identify all the information that is known or given. Check all that apply.

• 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

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. (Correct answer)
• D. Write an expression for any natural number, square it, and show it is divisible by 4.

Question 3

Let variables represent any whole number.
Then the expression <blank>\(2k\)</blank> represents any and all <blank>even</blank> numbers.