My fragile attempt: Note that if $1987^k-1$ ends with 1987 zeros, that means $1987^k$ has last digit 1 (and 1986 "next" ones are zeros). For this to be satisfied, $k$ has to be in form $k=4n$, where $n\in N$. This means out number can be written in form
$ [(1987^n)^2+1][1987^n+1][1987^n-1]. $
This number has to be dividable by $10^{1987}$ if there is such a number that is asked for in question.
Now, I believe that the fact 1987 is a prime is very important here. There are probably some theorems from number theory about primes and their powers. For example, if $p$ is a prime (distinct from 2 if needed), are there any important things about number such as $p^2-1$?
If I'm going at the right direction with this, I'd appreciate a hint. Please don't use too advanced techniques if possible. Thanks.