Can we assume that if for all values of , then for all ? Somehow I don't think this is valid, but let's run with it for a moment.
If this is true, then the whole point of the two-part exercise was to make the connection that for some where
Multiplying gives us . Now reduce the term and because the LHS is a whole number, the RHS must reduce fully. Since , 19 will still be a factor in this term after reducing, so it can be written for some positive integer k, which implies your hypothesis.
Not sure if this quasi-proof holds water, but it is the only connection I see between the two parts of the question.