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Lucky seven III (Posted on 2011-01-05) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
Rating: 4.5000 (2 votes)

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Solution without the extra challenge (spoiler) | Comment 1 of 4

[N/199] is

 3908431044109436069235064209938581797878280290340591848129536571747627024008933
55667225013958682300390843104410943606923506420993858179787828029034059184812953
65717476270240089335566722501395868230039084310441094360692350642099385817978782
80290340591848129536571747627024008933556672250139586823003908431044109436069235
06420993858179787828029034059184812953657174762702400893355667225013958682300390
84310441094360692350642099385817978782802903405918481295365717476270240089335566
72250139586823003908431044109436069235064209938581797878280290340591848129536571
74762702400893355667225013958682300390843104410943606923506420993858179787828029
03405918481295365717476270240089335566722501395868230039084310441094360692350642
09938581797878280290340591848129536571747627024008933556672250139586823003908431
04410943606923506420993858179787828029034059184812953657174762702400893355667225
01395868230039084310441094360692350642099385817978782802903405918481295365717476
27024008933556672250139586823003908431044109436069235064209938581797878280290340
59184812953657174762702400893355667225013958682300390843104410943606923506420993
85817978782802903405918481295365717476270240089335566722501395868230039084310441
09436069235064209938581797878280290340591848129536571747627024008933556672250139
58682300390843104410943606923506420993858179787828029034059184812953657174762702
40089335566722501395868230039084310441094360692350642099385817978782802903405918
48129536571747627024008933556672250139586823003908431044109436069235064209938581
79787828029034059184812953657174762702400893355667225013958682300390843104410943
60692350642099385817978782802903405918481295365717476270240089335566722501395868
23003908431044109436069235064209938581797878280290340591848129536571747627024008
93355667225013958682300390843104410943606923506420993858179787828029034059184812
95365717476270240089335566722501395868230039084310441094360692350642099385817978
78280290340591848129536571747627024008933556672250139586823003908431044109436069
235064209

The sum of its digits is 8745. The sum of those digits is 24, and those digits sum to 6, which is therefore the digital root.

 10   A=int(7*(10^2010-1)//9/199)
 20   print A
 30   Dr=A
 40   repeat
 50       Dr2=Dr
 55       Dr=0
 60       while Dr2>0
 70          Dr=Dr+Dr2 @ 10
 75          Dr2=Dr2\10
 80       wend
 85   print Dr
 90   until Dr<10


 


  Posted by Charlie on 2011-01-05 14:42:16
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