I got some problem with those demonstrations and I don't know where I'm wrong, let me show you my steps:
1: first of all
$ 6 | 2n(n^2 +2) $
That is, I must demonstrate that $6$ divides $2n(n^2 +2)$ , so I just simplify this stuff dividing by two:
$ 3 | n(n^2+2) $
to solve this I thought to model this with a simple modular algebra problem of the type:
$ n(n^2+2)≡0 ($mod $ 3)$
That is, I must try to satisfy this system of modular equations:
$ n≡0($mod $ 3)$
$ n^2≡-2($mod $ 3)$
but the amount of numbers that are smaller than 3 and co-prime with it are 2, and $gcd(2,2)=2$ meaning that we can't find the inverse of the application $x → x^2$.
2: Let $A:=Z/50Z$ and let $B:= (Z/50Z)*$ be the subset of A made of its invertible classes (mod 50). Calculate the cardinality of the following sets:
$ Y:=f∈A^A|f(B) =B $;
$ Z:=f∈Y|$ f is injective.
with this I don't even know where to start :(