N(k) = (10^k -1)/3 = 3*(10^k -1)/9
Since 499 is not divisible by 3, (it is prime in fact) there must be a rep unit of length k divisible by 499. (It will be the same k whether we look at a string of 3s or a string of 1s.)
I can't prove it, but I suspect the length of the repeating decimal will be 498.
So, I am guessing that a string of 498 3s is divisible by 499
Detour: look at a smaller number first. Consider what length rep unit is divisible by 13. I'll assume it is a string of 12 3s. (A string of 6 13s also works, but continue anyway ...).
Multiplying by a string of 3s is similar to multiplying by 1/3. So instead of looking at 333333333333/13, look at (1/3)*(1/13) = 1/39
= 0.02564102564102564
333333333333/13 = 25641025641
333333/13 = 25641
End of detour.
Back to our problem:
(1/3)*(1/499)= 1/1497 =
0.00066800267201068804275217100868403473613894455
57782231128924515698062792251169004676018704074
81629926519706078824315297261189044756179024716
09886439545758183032732130928523714094856379425
51770207080828323313293253173012692050768203072
81229124916499665998663994655978623914495657982
63193052772211088844355377421509686038744154976
61990647962591850367401469605878423513694054776
21910487641950567802271209084836339345357381429
52571810287241148964595858383433533734134936539
7461589846359385437541750167|
0006680026720106880427522
I have inserted a pipe (|) character after 498 digits after the decimal place, and this does correspond to where the decimal starts to repeat.
Multiplying (the 498 digits without the 2 leading zeros) 66800267201068804275217100868403473613894455577
82231128924515698062792251169004676018704074816
29926519706078824315297261189044756179024716098
86439545758183032732130928523714094856379425517
70207080828323313293253173012692050768203072812
29124916499665998663994655978623914495657982631
93052772211088844355377421509686038744154976619
90647962591850367401469605878423513694054776219
10487641950567802271209084836339345357381429525
71810287241148964595858383433533734134936539746
1589846359385437541750167
by 499 indeed results in a string of 498 3s.
So the last 5 digits are 50167
a different way
