A cryptographer is analyzing a security protocol that relies on the properties of prime numbers. They are required to prove the following theorem for any positive integer $n$:

“If $n^2$ is a multiple of 7, then $n$ must be a multiple of 7.”

The start of the proof by contradiction is shown in the box below.

Assumption: There exists an integer $n\in {\mathbb{Z}}^{+}$, such that $n^2$ is a multiple of 7, and $n$ is NOT a multiple of 7.

Case 1: Let $n=7k+1$ for some integer $k$.

$$n^2=(7k+1{)}^2=49k^2+14k+1=7(7k^2+2k)+1$$

which is not a multiple of 7.

Case 2: Let $n=7k+2$ for some integer $k$.

$$n^2=(7k+2{)}^2=49k^2+28k+4=7(7k^2+4k)+4$$

which is not a multiple of 7.

Case 3: Let $n=7k+3$ for some integer $k$.

$$n^2=(7k+3{)}^2=49k^2+42k+9=7(7k^2+6k+1)+2$$

which is not a multiple of 7.

Case 4: Let $n=7k+4$ for some integer $k$.

$$n^2=(7k+4{)}^2=49k^2+56k+16=7(7k^2+8k+2)+2$$

which is not a multiple of 7.