[Guest]

N is a 50-digit duodecimal (base 12) positive integer. All the digits of N is 1 except the 26th digit (reading left to right).

Reading left to right, what is the 26th digit of N, given that the duodecimal number 17 divides N?

[Guest]

Phew, think I got it!

7.

I did it with a pen and paper so the explanation is kind of long. Hopefully I can find time to write it out tonight. But in the meantime,

N/17 = 00832700832700832700832701008327008327008327008327 (all base 12 of course).

[superprismatic]

Tuckleton is correct.

The 26^th digit from the left is 7.

All those ones are just successive powers of 12 which can be taken mod 19 as you compute them. They all add up to 12 mod 19. The digit in the 26th place from the left is in the 12²⁴ place. Well 12²⁴=1 mod 19, so the digit we are looking for is 7 because 12(representing all the ones)+7*1(value of the 26^th place from left)=0 mod 19.

Edited November 11, 2009 by superprismatic

[Guest]

I never really did get the hang of modular arithmetic. What I did is just make a table of 17*(1 through B) and used that to help me make the transition to base 12.

17*1=17

17*2=32

17*3=49

17*4=64

17*5=7B

17*6=96

17*7=B1

17*8=108

17*9=123

17*A=13A

17*B=155

I started working backwards trying to find N/17. I knew the last digit had to be 7 in order to give a 1 at the end. Which left B for the next column so I needed to add 2 to it so the next number had to be 2 which, after carrying the 1, left 4 in the next column so the next number had to be 3 and so on. I found 008327*17=111111 so I could just tack that on to itself 4 times to get the first 24 1's. I noticed that 01 would give 17 for the next 2 digits (7 being 26th from the left) which left 24 digits left letting me do 008327 4 times to finish it off. Not particularly elegant I admit. But I guess it got the job done