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.

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.

Explanation:

Identify the given information

In this step, we analyze the first part of the problem, which asks to identify all known or given information.
The problem statement at the top says: "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."
From this statement, the given (known) information is:

1. We start with an even number \(a\) (so, "The starting number is even" is given).
2. The definition of an even number is given: "Every even number can be written in the form \(2k\) where \(k\) is any whole number."

The statements about the square of the starting number being even or divisible by 4 are what we want to prove, not what is initially given.
Therefore, the correct choices for the first part are A and B.
We apply the Even Number Definition and Mathematical Proofs concepts here.

Determine the proof process

In this step, we analyze the second part of the problem, which asks: "What process can be used to prove that the square of the number is divisible by 4?"
To prove that the square of an even number is divisible by 4:

1. We must represent any even number algebraically as \(2k\), where \(k\) is an integer/whole number.
2. We then square this expression: \((2k)^2 = 4k^2\).
3. Since \(4k^2\) is a multiple of 4, this shows the result is divisible by 4.

This matches option C: "Write an expression for any even number, square it, and show the result is divisible by 4."
We apply the Mathematical Proofs concept to select this logical path.

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.