N is a base ten positive integer formed by writing the digit 7 precisely 2010 times, that is N = 77....77 (2010 times).

Determine the digital root of [N/199].

Note: [x] denotes the greatest integer ≤ x.

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

N/199 is a large number but its digits will eventually repeat.  So I started doing the long division by hand (just kidding I cut and pasted from Charlies post) to find that it repeats this 100 digit string:

3908431044109436069235064209938581797878280290340591848129536571747627024008933
55667225013958682300

these digits sum to 430 (I did do that by hand) and so the digital sum of the string is 7.

[N/199] is a 2008 digit number so it repeats the string 20 times with 8 extra digits left over: 39084310 These numbers have digital root 1.

So the answer is 20*7+1 = 141
1+4+1 = 6

Posted by Jer on 2011-01-05 14:58:35