Is there something faulty about this statement? Show any prime of the form $3k+1$ is of the form $6k+1$. 
I came up with my own solution that made perfect sense to me, but when I read the text's solution, it argued that for the primes that are of the particular form are $6k+1 = 3(2k)+1$. But doesn't that really say the primes in the form of $3k+1$ are in the form of $6m+1$? It seems to me as though there's some misuse of notation here -- allowing $k = 2k$. So should the exercise be phrased as $6m+1$ instead? 

$3n+1=6k+1$ iff $n=2k.$ If $n$ is even, this will always work, but if $n$ is odd, then $3n+1$ is even and thus not prime and there are no solutions and so the statement is true.
You can of course extend this to the rational numbers, and then it gives every prime, but that's not very interesting.
