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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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Bigger numbers | Comment 6 of 27 |

I decided to check my math by evaluating this anyway.

27195^8 = 299166186312683900806365263000390625
10887^8 = 197363020619290527021005277168321
10152^8 = 112826956359514752722139774713856

N= 299081650248424125032066397497936160

N/37= 8083287844552003379245037770214490.810810810810810...

I didn't check the other factors of 26640 but they must have worked themselves out just fine as the repeating decimal is
810/999 = 30/37

30 was the remainder I predicted mod 37 but there was no guarantee that 2^4, 3^2, and 5 were going to leave no remainder.


  Posted by Jer on 2005-05-25 17:47:00
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