##ConnectedComponents
Using Daniel Lichtblau's answer to a related question
ConnectedComponents[pairs] //Sort /@ # & //Sort (* thanks: CarlWoll *)
{{3, 5, 9},
{11, 21, 22, 35},
{12, 14, 16, 23},
{1, 6, 10, 13, 36},
{17, 20, 24, 25, 28, 32},
{2, 8, 15, 18, 27, 29, 31},
{4, 7, 19, 26, 30, 33, 34}}
In versions prior to 10.3 use
ConnectedComponents[Graph[UndirectedEdge @@@ pairs]] //Sort /@ # & //Sort
##MatrixPower
Implementing transitive closure using MatrixPower:
m = Max@pairs;
(*the adjacency matrix of atomic elements in pairs:*)
SparseArray[pairs ~Append~ {i_, i_} -> 1, {m, m}];
(*symmetrize the adjacency matrix:*)
% + %\[Transpose] // Sign;
(*find the transitive closure:*)
Sign @ MatrixPower[N@%, m];
(*eliminate duplicate rows,and extract the atomic elements of pairs in each row:*)
Select[DeleteDuplicates @ Normal @ %, Tr@# > 1 &];
Join @@ Position[#, 1] & /@ %;
(*organize:*)
Sort[Sort /@ %]
{{3, 5, 9},
{11, 21, 22, 35},
{12, 14, 16, 23},
{1, 6, 10, 13, 36},
{17, 20, 24, 25, 28, 32},
{2, 8, 15, 18, 27, 29, 31},
{4, 7, 19, 26, 30, 33, 34}}