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Fifth powers (Posted on 2013-01-06) Difficulty: 3 of 5
Given that n5 = 1335 + 1105 + 845 + 275.
Find that value of n using only pen and paper.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Assuming... | Comment 1 of 8
Assuming n is an integer (why else would you ask for it?) we can narrow it down rather quickly.
Clearly n>133
A simple upper bound is 133*4^(1/5) < 133*1.5 < 200
(I could probably do better but this is fine.)
So 133<n<200

Now to check factors.
I'll spare the details but using modular arithmetic one can easily show 2, 4, 8, 16 are factors but not 32.
3 and 9 are both factors but not 27.

There's only one number in the interval that is a factor of both 16 and 9: 16*9 = 144.

I'll admit I then used a calculator to prove this works (though it's not hard to do by hand, just tedious).  Both sides equal 61917364224.  So the assumption n is an integer was correct.

Edited on January 7, 2013, 12:53 pm
  Posted by Jer on 2013-01-06 19:04:30

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