Elements in a finite quotient group

Question

The question is as follows:

Suppose that $K\mathrel{\unlhd}G$ with $|G/K|=n<\infty$.

1. Then $g^n\in K~\forall~g\in G$.
2. If $g\in G$ and $g^m\in K$ for some integer $m$ such that $\text{gcd}(m,n)=1$ then $g\in K$.

My attempt:

I have successfully proved 1, using the fact that $|G/K|=n\Rightarrow (gK)^n=K\Rightarrow g^nK=K\Rightarrow g^n\in K.$

Now, in the second part, I get that $g^n, g^m \in K$ so, $g^ng^m\in K$. Now, I was wondering if we could somehow use the fact $\exists~x,y\ni xm+yn=1.$ I'd appreciate some help in this part of the question.