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Strange divisibility (Posted on 2005-05-25) Difficulty: 2 of 5
Without evaluation of it, prove that the number N = 27,195^8 - 10,887^8 + 10,152^8 is divisible by 26,460.

Note: the original problem mistakenly listed the last number as 26,640. This has been corrected

See The Solution Submitted by pcbouhid    
Rating: 3.6667 (3 votes)

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re(5): Remove Problem | Comment 18 of 27 |
(In reply to re(4): Remove Problem by pcbouhid)

Taking the 2nd, 4th and 8th powers, mod the divisibility base, of each number separately, and adding and subtracting as was done here in fact does not evaluate the number N.  The highest number used in the calculation was  557,715,456, which was 23,616^2, before reduction mod 26,640. Nowhere was the value of N (299081650248424125032066397497936160 per Jer's post) computed in that solution.
  Posted by Charlie on 2005-05-27 15:20:26

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