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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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What is strange | Comment 22 of 27 |
What is strange about the divisibility in this problem is that the (correct) divisor 26,460 satisfies

26,460 = 27,195 - 10,887 + 10,152.

Thus, in this case, x^8 -y^8 + z^8 is divisible by x - y + z. However, x - y + z cannot be a factor of x^8 -y^8 + z^8 because there are many cases where x - y + z has the value 0 but x^8 -y^8 + z^8 does not.
  Posted by Richard on 2005-05-27 18:40:47
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