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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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Smaller numbers | Comment 5 of 27 |

27195 and 26640 are 0 mod 37

10887 is 9 mod 37
10152 is 14 mod 37

9^8 = ((9^2)^2)^2 = (7^2)^2 = 12^2 = 33 mod 37
14^8 = ((14^2)^2)^2 = (11^2)^2 = 10^2 = 26 mod 37

0 - 33 + 26 = -7 = 30 mod 37 so N is not divisible by 26640

 


  Posted by Jer on 2005-05-25 17:30:49
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