An integer c is square-friendly if it has the following property: For every integer m, the number m^{2}+18 m+c is a perfect square. (A perfect square is a number of the form n^{2}, where n is an integer. For example, 49=7^{2} is a perfect square while 46 is not a perfect square. Further, as an example, 6 is not square-friendly because for m=2, we have (2)^{2}+(18)(2)+6=46, and 46 is not a perfect square.)
In fact, exactly one square-friendly integer exists. Show that this is the case by doing the following:
(a) Find a square-friendly integer, and prove that it is square-friendly.
(b) Prove that there cannot be two different square-friendly integers.