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
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.

Explanation:

Analyze the given statement

The problem statement at the top reads: "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."

Identify the starting assumptions

In a direct proof of the statement "the square of an even number is divisible by 4", we begin by assuming the hypothesis.

• The hypothesis is that we start with an even whole number, \(a\). Thus, "The starting number is even" is a known/given assumption.
• We are also given the definition of an even number: "A whole number \(a\) is even if and only if there is a whole number \(k\) such that \(a = 2k\)." This means every even number can be written in the form \(2k\) where \(k\) is a whole number.

Evaluate the goal statements

The statements "The square of the starting number is even" and "The square of the starting number is divisible by 4" are parts of what we want to prove (the conclusions), not information that is already known or given at the start of the proof. Therefore, only the starting assumptions are the given information.

Answer:

• 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.