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
If we square a number, square the result and square that result, all mod 26,640, the result will be the eighth power of the number mod 26,640.
Using that, the 2nd, 4th and 8th powers of the three base numbers involved are:
2nd 4th 8th
27,195 14,985 1,665 1,665
10,887 5,409 6,561 23,121
10,152 19,584 23,616 7,056
But taking 1,665  23,121 + 7,056 results in 14,400, which is not congruent to zero mod 26,640. So n is not divisible by 26,640.

Posted by Charlie
on 20050525 13:24:34 