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Divisibility of 29 (Posted on 2013-05-15) Difficulty: 3 of 5
The integers x,y,z are such that 29 divides the sum x4+y4+z4. Determine if 294 also divides x4+y4+z4.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 4.0000 (1 votes)

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Solution Rest of solution | Comment 2 of 4 |
The question is whether x^4+y^4+z^4 could be a multiple of 29 but not all of x,y,z multiples of 29.

Consider the residuals (mod 29) of n^4 for n=0,1,2,...,14
0,1,16,23,24,26,20,23,7,7,24,25,1,25,20

The surprising thing about this set is that no three of them sum to 0 (mod 29) except for 0+0+0

In other words, the solutions given in my other post are all of the solutions.  The determination is, therefore: YES.  If 29 divides the sum, then 29^4 will always divide the sum.

  Posted by Jer on 2013-05-15 12:40:39
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