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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 Analytic Answer Comment 4 of 4 |
N mod 9 is easy to compute: N mod 9 = 2010*7 mod 9 = 21*7 mod 9 = 3.

N mod 199 is trickier without a computer.  First express N as a sum:
N = Sigma{i=0 to 699} 777*1000^i

777 mod 199 = 180, 1000 mod 199 = 5.  Then:
N = Sigma{i=0 to 669} 180*5^i mod 199

The closed form of this sum is:
N = 180*(5^670 - 1)/4 mod 199

Since 199 is prime Fermat's Little Theorem implies x^198 = 1 mod 199.  With 670 = 3*198+76, this implies:
N = 45*(5^76-1) mod 199

This is small enough to evaluate by hand.  Then N = 186 mod 199.

From the Chinese Remainder Theorem:  N mod (9*199) = 199*3 + 9*87 = 1380.  

Then floor[N/199] mod 9 = floor[1380/199] mod 9
1380 = 6*199 + 186 and 199 mod 9 = 1.  Therefore floor[N/199] mod 9 = 6*1 = 6.

  Posted by Brian Smith on 2016-07-04 11:56:23
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