Showing that a function is well defined

Question

Let $F$ be a free abelian group.

Show that $\Bbb Z/p\Bbb Z \times F / p F \rightarrow F/pF: (a,v) \mapsto av$, where $av=(c+p\Bbb Z)(h+pF) = ch+pF$ is well defined.

I know this means to show that If $c+p\Bbb Z=c'+p\Bbb Z$ and $h+pF=h'+pF$, then $ch+pF = c'h'+pF$. I can see that, assuming the two arguments are equal, $c'=c+pj$ for some $j \in \Bbb Z $ and $h'=h+pg$ for some $g \in F$, but from here I'm stuck.

Anyone have any ideas?

Answer 1

... so $$ c'h'=(c+pj)(h+pg)=ch+pjh+p^2jg+cpg=ch+p\cdot(jh+pjg+cg)$$