I'm confused on how to solve this problem though without just "knowing" what hydroxyapatite is.

Big-Daddy nailed it (I mean the method, he was not the first to post the answer).

You are told it is a basic (OH

^{-}) calcium (Ca

^{2+}) phosphate (PO

_{4}^{3-}). Basic means no chances of HPO

_{4}^{-} and H

_{2}PO

_{4}^{-}. Molecule is Ca

_{k}(PO

_{4})

_{l}(OH)

_{m}. Then it is just a matter of writing correct charge balance (molecule must be neutral) and molar mass equations:

2k - 3l - m = 0

40k + 95l + 17m = 502

solving them in terms of a parameter and poking the solution space with small integers (which is just a fancy way of calling trial and error

).

For example you can solve these equations for k:

[tex]k = \frac{502-44l}{74}[/tex]

and plug l=1, 2, 3... into the equation. When l is 3, you get a nice integer value of 5 for k, so you just plug l=3 and k=5 into original set of equations and you get m=1 - so Ca

_{5}(PO

_{4})

_{3}OH is the correct answer.

In general this kind of a problem doesn't have to have a unique solution.