SOLUTION: Prove that n^8 − n^4 is divisible by 5 for any natural n.

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Question 1033741: Prove that n^8 − n^4 is divisible by 5 for any natural n.
Found 2 solutions by robertb, ikleyn:
Answer by robertb(5830)   (Show Source): You can put this solution on YOUR website!
N.B.: The proof given by the other tutor above is essentially the same proof here, making use of modular arithmetic, but is just much wordier and lacks further originality and insight.
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Proof:
If n is divisible by 5, then it is quite clear that is divisible by 5. Hence check for the other equivalence classes 5k + 1, 5k + 2, 5k + 3, 5k + 4.
It is easy to see that



.
Furthermore,



.
Hence,



,
and the statement is proved for all natural n.






Answer by ikleyn(52748)   (Show Source): You can put this solution on YOUR website!
.
Prove that n^8 - n^4 is divisible by 5 for any natural n.
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Below is another solution to the same problem.

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Let us factor  as far as we can:


 =  =  = 


Now, if n is a multiple of 5, then    is a multiple of 5.


If n gives a remainder 1 when divided by 5, then the factor (n-1) is a multiple of 5.


If n gives a remainder 4 when divided by 5, then the factor  (n+1)  is a multiple of 5.


If n gives a remainder 2 or 3 when divided by 5, then the factor   is a multiple of 5.


So, in any case    is a multiple of 5,  and the statement is proved.


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