N is a duodecimal (base 12) positive integer having precisely 50 digits such that each of its digits is equal to 1 except the 26th digit. If N is divisible by the duodecimal number 17, then find the digit in the 26th place.
The definition of -th place is ambiguous - one may count either from L to R or the other way .
Rather than discuss what should be the right assumption- let's solve it twice.
N= S(50 ones in a row)+(k-1)*12^deg deg =24 or 25
To evaluate the values of powers of 12, it it is easy if you notice the period, i.e. 6.
12^24 mod19 equals 12^0 mod19=1
12^25 mod19 equals 12^1 mod19=12
12^50 mod19 equals 12^2 mod19=11
S(50 ones in a row)mod 19=(((12^50 )-1)/(12-1))=13
N=13+(K-1)*R R =EITHER 1 OR 12 (ALL CALCULATED MOD19)
1st ans: k-1=6 since 13+6=19 =0 mod19 k=7
2nd ans: k-1=10 since 13+120=19*7 =0 mod19 k=11
So it is either 7(seven) or B(eleven).
Funny as it is - gamblers will like the solution.
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