A base ten number formed by writing the digit seven precisely 2011 times is denoted by N. That is:

N = 77……77 (2011 times).

Let us denote x = [N/9], y = [x/9] and, z = [y/9]

Determine the digital root of z.

Note: [P] denotes the greatest integer ≤ P

**** For an extra challenge, solve this puzzle without using a computer program.

The digital root of z is 6.

Lines 60 through 290 of the below program are not really needed for this solution, but were added to display the digital roots also for N, x and y, which are 1, 4 and 7 respectively.

list
   20   for I=1 to 2011:N=N*10+7:next
   30   X=N\9
   40   Y=X\9
   50   Z=Y\9
   60   Dr=0:Num=N
   70   while Num>0
   80        Dr=(Dr+Num @ 10) @ 9:Num=Num\10
   90   wend:print Dr
  160   Dr=0:Num=X
  170   while Num>0
  180        Dr=(Dr+Num @ 10) @ 9:Num=Num\10
  190   wend:print Dr
  260   Dr=0:Num=Y
  270   while Num>0
  280        Dr=(Dr+Num @ 10) @ 9:Num=Num\10
  290   wend:print Dr
  360   Dr=0:Num=Z
  370   while Num>0
  380        Dr=(Dr+Num @ 10) @ 9:Num=Num\10
  390   wend:print Dr
OK
run
 1
 4
 7
 6
OK

Posted by Charlie on 2011-09-13 13:41:33