Is n4 - 1 divisible by 5 when n is not divisible by 5?

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The discussion centers on proving that n^4 - 1 is divisible by 5 when n is not divisible by 5. Participants suggest using proof by cases, specifically considering values of n in the forms 5k + 1, 5k + 2, 5k + 3, and 5k + 4. Alternative forms like 5k - 2 and 5k - 1 are also proposed for simplification. The key insight is that the divisibility hinges on the expression m^4 being congruent to 1 modulo 5, which can be examined using the binomial theorem. This approach aims to clarify the conditions under which the original statement holds true.
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Homework Statement



Show n4 - 1 is divisible by 5 when n is not divisible by 5.

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The Attempt at a Solution



My thought here is proof by cases, one case being where n is divisible by 5 and the other case saying n is not divisible by 5. I am just not totally sure how to implement this strategy.
 
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If you split your second case a bit more that will do fine.
 
If n is not divisible by 5, then it is of the form n= 5k+ 1, n= 5k+ 2, n= 5k+ 3, or n= 5k+ 4.

Or you can consider n= 5k- 2, n= 5k- 1, n= 5k+ 1, and n= 5k+ 2 if that makes the calculations simpler.
 
HallsofIvy said:
If n is not divisible by 5, then it is of the form n= 5k+ 1, n= 5k+ 2, n= 5k+ 3, or n= 5k+ 4.

Or you can consider n= 5k- 2, n= 5k- 1, n= 5k+ 1, and n= 5k+ 2 if that makes the calculations simpler.

If we would to spent a little more time to think through , we will realize that the determining factor will then lies in 5k \pm m such the m^{4}\equiv 1 (mod5) with application with binomial theorem.
 

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