eaSteps[{a_, b_}] :=
  NestWhileList[{#[[2]], Mod[#[[1]], #[[2]]]}&, {a, b}, #[[2]] != 0 &]

eaSteps[{1736, 1333}]

{{1736,1333},{1333,403},{403,124},{124,31},{31,0}}`

eaData =
   SortBy[
     Flatten[
       Table[
         {a, b, Length[eaSteps[{a, b}]] - 1},
         {a, 1, 100},
         {b, 1, a}],
       1],
     Part[#,3]&]

ClearAll[maxEALengthPair]
maxEALengthPair[n_] := Module[{i = 0}, While[Fibonacci[++i] < n]; Fibonacci[{i-1, i-2}]];

TeXForm @ Grid[{#, ## & @@ #2} & @@@ Transpose[{{"n",  "pair"},
 {#, maxEALengthPair /@ #} &@{50, 100, 500, 1000, 10000, 100000, 1000000, 10^7}}],
   Dividers -> All]

nl = {50, 100, 500, 1000} ;
{#, MaximalBy[Reverse /@ Subsets[Range[#], {2}], Length @* eaSteps][[1]] & /@ #}&@nl
Grid[{#, ## & @@ #2} & @@@ Transpose[{{"n",  "pair"}, {#,
   MaximalBy[Reverse /@ Subsets[Range[#], {2}], Length[eaSteps[#]] &][[1]] & /@ #} &@nl}],
  Dividers -> All]

MaximalBy[Reverse /@ Subsets[Range[50], {2}], Length @* eaSteps ]

Length[eaSteps @ #] - 1 &/@ %

n (n + 1)/2

Length[Subsets[Range[n], {1, 2}]]

100 101 /2

Length[Subsets[Range[100], {1, 2}]]

table[n_Integer] := Flatten[Table[{a, b, foo}, {a, 1, n}, {b, 1, a}], 1];
Length[table[100]]

MaximalBy[Reverse/@Subsets[Range[1000],{2}], Length[eaSteps[#]]&]

Length[Length[eaSteps[{987, 610}]]] -1