2002 AIME I Problem 9

Harold, Tanya, and Ulysses paint a very long picket fence. Harold starts with the first picket and paints every hth picket; Tanya starts with the second picket and paints every tth picket; and Ulysses starts with the third picket and paints every uth picket. Call the positive integer 100h+10t+u \textit{paintable} when the triple (h,t,u) of positive integers results in every picket being painted exactly once. Find the sum of all the paintable integers.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.