9

Part of 2002 AIME Problems

Problems(2)

Source:

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 everth

t

th picket; and Ulysses starts with the third picket and paints every

u

th picket. Call the positive integer

100h+10t+u

<span class='latex-italic'>paintable</span>

when the triple

(h,t,u)

of positive integers results in every picket being painted exaclty once. Find the sum of all the paintable integers.

number theoryrelatively prime

Counting Disjoint Subsets

Source:

\mathcal{S}

\{1,2,3,\ldots,10\}.

n

be the number of sets of two non-empty disjoint subsets of

\mathcal{S}.

(Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when

n

1000.

countingdistinguishability