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.

If the digits were all 1's, the value of the number, in decimal would be (12^50 - 1)/11. When taken mod 19 decimal that comes out as 13 decimal.

The 25th position has place value 12^24, which, mod 19, is 1

So we need to add another 19-13 = 6 to that position, making it a 7.

The calculations were done in immediate UBASIC:

?(((12^50)-1)//11)@19
 13
OK
?(12^24)@19
 1

 

Posted by Charlie on 2010-05-29 15:57:08