A researcher investigating number theory properties is formalising a proof for the theorem:

“For any positive integer $m$, if $m^2$ is divisible by 7, then $m$ must be divisible by 7.”

The initial steps of the proof by contradiction are provided below.

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

Case 1: Let $m=7k+1$ for some integer $k\ge 0$.

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

which is not a multiple of 7.

Case 2: Let $m=7k+2$ for some integer $k\ge 0$.

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

which is not a multiple of 7.

Case 3: Let $m=7k+3$ for some integer $k\ge 0$.

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

which is not a multiple of 7.