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

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