Reachability and cheap calculation

mahch

I run into the following question:
Given p,q element of {7,11,13,17,19,23,29,31} and also given that :
x,y is an element of N+{0}.

The search is for a formula for "z" as element of N whereby "z" is represented by :

z = px + qy + 30xy.

The formula(tion) should tell instantly whether z is reachable or not for any z element N and provide for a "cheap calculation" of the value of "z"