1. Define the Variables

Let the digits of $M$ be $a$ and $b$.
- $\(M=10 a+b\)$
- $\(N=10 b+a\)$ (the digits reversed)

Here, $a$ and $b$ are integers from 0 to 9 , and since $M$ is a two-digit number, $a \neq 0$.
2. Set up the Expression for $K$

Given $\(K=M+N\)$ :

$$
\(\begin{gathered}
K=(10 a+b)+(10 b+a) \\
K=11 a+11 b \\
K=11(a+b)
\end{gathered}\)
$$

3. Evaluate the Statements
- A. K must be even: Not necessarily. If $a=1, b=2$, then $K=11(3)=33$, which is odd.
- B. K cannot be square: False. If $a+b=11$ (e.g., $M=29, N=92$ ), then $K= 11(11)=121$, which is $\(11^2\)$.
- $\(\mathrm{C} . \mathrm{K}\)$ cannot be divisible by $\(\mathbf{1 3}\)$ : False. If $a+b=13$ (e.g., $M=49, N=94$ ), then $K=11(13)$, which is divisible by 13 .
- D. K must be divisible by 11: True. Our simplified expression $K=11(a+b)$ shows that 11 is always a factor of $K$.
- $\(\mathbf{E}\)$. If $\mathbf{M}$ is even then $\(\mathbf{K}\)$ must be even: False. If $M=12$ (even), $N=21 . K=12+ 21=33$, which is odd.

Correct Answer: D. K must be divisible by 11