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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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Question Hmmmm..... (spoiler?) | Comment 1 of 27

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 2005-05-25 13:24:34
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