Discrete math, statements true or false 
The universal set defined for the task is $\{2,3,4,5,\dots\}$.
I've been tearing my hair out at this one for quite some time. 
I can make the top one true, as in my head you can always just select $n=m$ and $k=1$.
However, the correct answer is apparently 2), 3) and 4). I can recognize 2) as Fermat's (?), but beyond that I reckon I'm thinking myself more confused than not. 
Any explanations to a lad with an exam in a few days? :-)

For the first problem, you are not allowed to use $k=1$, it is not in the universe. So any $n$ that is not a perfect power with exponent $\gt 1$ is a counterexample. The first few are $2,3,5,6,7,10$. 
For the third problem, you can take $n=2$. With that choice, the sentence is true, since $m$ and $m^n$ have the same parity. 
For the fourth problem, take $k=2$ and $n=5$.
And you are right about the second problem. Given $m\ge 2$, we can find a prime $n$ such that $n$ does not divide $m$. For that prime, the congruence will hold.
