$A,C$ are recursive and $A\cdot B = C$, Is $B$ is recursive?

Question

Past year paper question.

Prove or disprove: Suppose $A,C$ are recursive and $A\cdot B = C$ (The dot denotes concatenation). Then $B$ is recursive.

I think it is false.

Let $B$ be any non-recursive language containing the empty string $\epsilon$.

Let $A=C=\Sigma^*$.

Then $A \cdot B = C$.

Question is how can I construct $B$?

Answer 1

Probably you can just suppose that there exists a non-recursive language that contains the empty string, and nobody will demand a proof of this.

Adding or removing a finite number of strings will not change the recursiveness or non-recursiveness of a language. In particular, adding the empty string will not change the complexity (take a TM for the language, add by a non-deterministic choice the possibility to accept the empty string instead of starting the original computation; even if the original TM rejects the empty string, the new one will accept it). So if you are allowed to suppose without proof that there exist non-recursive languages, then you can just take any of them and add the empty string and it will still be non-recursive.