Proof by Contraposition... (and background hypothesis PLEASE HELP!)

Prove by using proof by contraposition (and background hypothesis) that "the product of a non-zero rational number m and irrational number n is irrational....

im supposed to use contraposition and background hypothesis to change this statement to...

in the background ---> (m not equal to 0): and the proof ---> (if mn is rational, then n is rational)

BUT i do not know where/if im supposed to incorporate the "background" portion of my hypothesis into my proof (because i do know that you cant prove "if mn is rational then n is rational" since such a proof does not exist.)

You said yourself that without the hypothesis that m <> 0 you can't prove the claim, so you have to use it. Is there any reason not to?

Here is an outline: you are proving a statement of the form A -> (B -> (C -> D)) where A is "m is rational", B is "m <> 0", C is "n is irrational" and D is "mn is irrational". This is equivalent to A -> (B -> (~D -> ~C)). In proving the latter, you assume A, B and ~D and prove ~C.

To explain the formula A -> (B -> (C -> D)): this is equivalent to A /\ B /\ C -> D. Indeed, saying "if A, then: if B, then C" is the same as saying "if A and B, then C". Formally this can be seen using truth tables.