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.
well if x is the missing digit then the base 10 representation of N is
N = x*12^25 + sum(12^t, t=0 to 24) + sum(12^t, t=26 to 49)
using a calculator and the equation for geometric sums we get
N = 953962166440690129601298432x + 82731255909110452484796228821879345604102166791943261
now 17 base 12 is 1*12+7=19 base 10
so we can take the above equation mod 19 to get
N = 12x + 1
so we want to know for what x in [0,11] is 12x+1 divisible by 19
this is small enough field for guess and check which gives the only answer as x=11 or D in duodecimal
thus we have
N = 111111111111111111111111D1111111111111111111111111
in base 12 or
82731255909110452484796239315463176451693592406226013
in base 10

Posted by Daniel
on 20100529 13:37:30 