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Duodecimal Digital Expansion Puzzle (Posted on 2023-03-11) Difficulty: 3 of 5
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.

See The Solution Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

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Solution Analytic Solution Comment 3 of 3 |
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).

  Posted by Brian Smith on 2023-03-11 14:56:45
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