Consider the digital expansion of this duodecimal fraction:
(4)12
--------
(19)12
Determine the (419)
12th duodecimal digit to the right of the duodecimal point.
Notational notes: 0.x[yz] denotes the duodecimal expansion with a fixed x and repeating block yz, thus 0.x[yz] = 0.xyzyzyzyz.....
19(base12) factors into 3*7, this is important because 3 is a factor of the base 12. So our first step is to decompose the fraction into two fractions, one with denominator 3 (divisor of the base) and one with denominator 7 (coprime to the base). This is pretty easy as we quickly get 4/19(base12) = 1/3(base12) - 1/7(base12).
1/3(base12) is expressible as finite duodecimal 0.4(base12). Or the repeating B's form is 0.3[B](base12)
1/7(base12) is expressible as a repeating duodecimal. The length of the repetition is some factor of 7-1=6. So we'll just take the length to be 6. Then 1/7(base12) = ??????/BBBBBB(base12); which then makes the repeating duodecimal ?????? block equal BBBBBB/7(base12) = 186A35(base12).
Then 4/19(base12) = 1/3(base12) - 1/7(base12) expanded into duodecimal is 0.3[BBBBBB](base12) - 0.1[86A351](base12) = 0.2[35186A](base12).
So now just to determine the 419th(base12) digit in the expansion. 419(base12) mod 6 = 3. The 3rd(base12) digit in the duodecimal expansion is 5 and because the expansion repeats every 6 digits after the first digit, then the 419th(base12) digit is also 5.
An aside: I see both Charlie and Larry have submitted computer program solutions, which I believe circumvents the whole point of this puzzle: to take the well-known concept of repeating decimals and work out the details of how to apply that concepts to an unfamiliar base representation (base 12 in this case).
Analytic Solution
