How do I prove that 5 divides x^5 - x?

Thread starter ninjagod123
Start date Feb 10, 2010

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SUMMARY

The discussion focuses on proving that 5 divides the expression x⁵ - x. Participants suggest computing the fifth powers of integers modulo 5, specifically for x = 0, 1, 2, 3, and 4. It is concluded that checking only x = 0, 1, and 2 is sufficient due to the properties of modular arithmetic, where 4 is equivalent to -1 and 3 is equivalent to -2 modulo 5.

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Feb 10, 2010

ninjagod123

How do I prove that 5 divides x^5 - x??

How do I prove that 5 divides x^5 - x??

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Feb 10, 2010

CRGreathouse

Science Advisor

Homework Helper

Compute the fifth powers mod 5.

Feb 10, 2010

CompuChip

Science Advisor

Homework Helper

Let me give you a hint: checking it for x = 0, 1, 2, 3 and 4 suffices.

(In fact, as 4 = -1 (mod 5) and 3 = -2 (mod 5), you can see that only x = 0, 1, 2 will do).

I'll let you figure out why.