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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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Some Thoughts Solutions I can find (partial solution) | Comment 1 of 4
Obvious solutions
x=y=z=0

If x=y=z=29 then the sum is 3*29^4 which is divisible by 29^4.

You can generalize this result to:

(29a)^4 + (29b)^4 + (29c)^4 = 29^4(a^4+b^4+c^4) for any integers a,b,c.

So the answer to the problem is either "sometimes" or "yes, always"

Are there solutions where x,y,z are not all multiples of 29?

  Posted by Jer on 2013-05-15 12:25:00
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