We\'re going over formal proofs. I\'m not sure if I should account for separate

Question

We're going over formal proofs. I'm not sure if I should account for separate cases where k is even or odd or not.

I would like to see a formal proof done to see if I'm on the right track.

Suppose that we call an integer "threeven" iff it can be written as 3k for some integer k. Suppose that we also say that an integer is "throdd" iff it can be written in the form 3k+1 for some integer k, and further say that an integer is "double-throdd" iff it can be written in the form 3k+2 for some integer k.       

Similar to what happens with good old-fashioned even and odd, the square of any threeven integer is threeven, and the square of any throdd integer is throdd.   In this question you will need to determine what happens when you square a double-throdd integer.       

Only one of the following conjectures is true.   Determine which conjecture is true, and prove it.

Explanation / Answer

let x= 3k+2   (double-throdd)

x^2 =9k^2 + 12k + 4

x^2 =9k^2 + 12k + 3+1

x^2 =3(3k^2 + 4k + 1) + 1

let p=3k^2 + 4k + 1

3^2 =3p + 1 (throdd)

The square of any double-throdd integer is throdd.

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