There are 5 "prime months" (2, 3, 5, 7, 11)
For the dates, there are 12 (2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, 31)
So there are \(5 \times 12 = 60\), but we overcount for 2/29, 2/31, and 11/31, so there are \(60 - 3 = \color{brown}\boxed{57}\) "prime dates".