Question 9
Harold, Tanya, and Ulysses paint a very long picket fence.
• Harold starts with the first picket and paints every $h$th picket;
• Tanya starts with the second picket and paints every $t$th picket; and
• Ulysses starts with the third picket and paints every $u$th picket.
Call the positive integer $100h + 10t + u$ *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.
• Harold starts with the first picket and paints every $h$th picket;
• Tanya starts with the second picket and paints every $t$th picket; and
• Ulysses starts with the third picket and paints every $u$th picket.
Call the positive integer $100h + 10t + u$ *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.