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Consecutive Conclusion (Posted on 2014-09-30) Difficulty: 3 of 5
Determine the maximum possible value of N for which 311 is expressible as the sum of N consecutive positive integers.

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (2 votes)

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Solution A higher N (spoiler) | Comment 2 of 8 |
The sum of N consecutive positive integers, the first of which is a, is (a + (a+N-1))N/2.

If this equals 3^11, then either N or N/2 is a factor of 3^11.

Thus, N can only be 3^k where k is between 0 and 11, or 2*3^k where k is between 0 or 10.

But some of these are obviously too large, since N^2 < 2*3^11.
From this equation, N < sqrt(6) * 3^5

Of our candidates, the largest possible N is 2*3^5.

Will this work?
Substituting and solving for a, 
  (2a+ 2*3^5 - 1)*3^5 = 3^11
  a = (3^6 + 1 - 2*3^5)/2 = (3^5 + 1)/2

This is a whole number, so the maximum N is indeed 2*3^5.

The last term is (5*3^5 - 1)/2.
The average term is 3^6/2, so the total sum is indeed 3^11

  Posted by Steve Herman on 2014-09-30 14:46:10
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