Let A=(0, a), O=(0,0), C=(c, 0), B=(c, b), where a, b, c are positive integers. Let P=(p, 0) be the point on line segment O C that minimizes the distance A P+P B, over all choices of P. Let X=A P+P B.
(a) Show that this minimum distance is X=\sqrt{c^{2}+(a+b)^{2}}
(b) If c=12, find all pairs (a, b) for which a, b, p, and X are positive integers.
(c) If a, b, p, X are all positive integers, prove that there exists an integer n \geq 3 that divides both a and b.