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50 - Digit Number II (Posted on 2010-05-29) Difficulty: 2 of 5
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.

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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Solution For gamblers and shoppers | Comment 7 of 9 |

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.

 

 

9

 

 

 

9


  Posted by Ady TZIDON on 2010-05-29 17:25:47
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