I think our large number is divisible by 81.

Consider the fractions you get from 1/3^n

Start with 1/27: 0.037037037037037...     it has period 3

Take a number of iterations of 037, say 5 iterations.

037037037037037  which is 15 digits

Drop the leading zero and multiply by 9 (one less power of 3 than our fraction). 

Result:   333333333333333 which is 15 digits

This should work for any length which is a multiple of 3.  They should all be divisible by 9.

What about 1/81:  .0123456790123456789 etc  period 10

but multiplying this by 27 yields strings of 17 3s followed by 1 zero.

What about 1/243:  (thanks to full precision calculator)

0.004115226337448559670781893

004115226337448559670781893

etc with period 27

4115226337448559670781893*81 =

    333333333333333333333333333  (27 digits)

But if you do multiple copies of the 27 digits, 004115226337448559670781893, delete the 2 leading zeros and multiply by 81 you get a string of 3s of length 27k where k is the number of copies.

So for any length of 3s that is a multiple of 27, it should be divisible by 81.

Is 3^2013 a multiple of 27?  

Yes 3^2013 = 27 * 3^2010

This procedure might be continued, but I cannot explain why it worked for 1/27 and 1/243 but not 1/81.

Maybe it works on every other power.  Maybe it only works if the repeating decimal has a period which is a multiple of 3. ??

More testing needed.