PROBLEM STATEMENT
Find all functions $f: \mathbb N \rightarrow \mathbb N$ which satisfy:
(a) $ f $ is a surjective function;
(b) $m|n$ if and only if $f(m)|f(n)$, for any two natural numbers $m, n$. $ \ \ \ \ $ $(1)$
MY PROGRESS
I am unable to make progress in this question. All I could figure out is $f(1)=1$:
Since $f$ is surjective, there exists $a \in \mathbb N$ such that $f(a)=1$. Now, replace $n$ by $a$ and $m$ by $1$ in $(1)$ to obtain $$f(1)|f(a) \implies f(1)|1 \implies f(1)=1 .$$This follows from the fact that $1|a$ for any choice of $a$.
Alternatively:
Replacing $m$ by $a$ and $n$ by $1$, we obtain $$a|1 \iff f(a)|f(1) , $$ from here we conclude that $f(1)=1$.
Are both the arguments above are correct? I am confused, as I think only one is correct.
How should I progress further? Any ideas or hints will be much appreciated.
Thanks.
NOTE: I have edited this question from understanding-solution to this.