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

Thread starter ninjagod123
Start date Feb 10, 2010

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To prove that 5 divides x^5 - x, one can compute the fifth powers of integers modulo 5. Checking the values for x = 0, 1, 2, 3, and 4 shows that only x = 0, 1, and 2 need to be considered. This is because 4 is equivalent to -1 and 3 is equivalent to -2 in modulo 5 arithmetic. The resulting calculations confirm that x^5 - x is divisible by 5 for these values. Thus, it is established that 5 divides x^5 - x for all integers x.

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.

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