My problem is as follows:
Let $c,d,n$ be fixed natural numbers with $c<d$. I am looking for an injection $f$ from $\mathbb{N}$ to $\mathbb{N}$ so that $\forall a\le n$ and $b\in\mathbb{N}$, we have $\{f(ab),f(ab+a)\}=\{a_1c+a_1b_1,a_1d+a_1b_1\}$ for some natural numbers $a_1,b_1$ with $a_1\le n$.
I have tried directly thinking of such a function but to no avail. How should I proceed?
Thanks.
