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

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In summary, divisibility is the concept of one number being evenly divisible by another number, with a remainder of 0 when divided. To prove this, one can use methods such as long division, prime factorization, or divisibility rules. When proving that 5 divides x^5 - x, it can be shown through methods such as mathematical induction, the binomial theorem, modular arithmetic, or polynomial division. These methods may vary based on individual knowledge and preference.
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ninjagod123
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How do I prove that 5 divides x^5 - x??

How do I prove that 5 divides x^5 - x??
 
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  • #2


Compute the fifth powers mod 5.
 
  • #3


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.
 

1. What is the definition of divisibility?

Divisibility is the mathematical concept of one number being evenly divisible by another number, meaning that when the first number is divided by the second number, the remainder is equal to 0.

2. How can I prove that a number is divisible by another number?

To prove that a number is divisible by another number, you can use a variety of methods such as long division, prime factorization, or the divisibility rules for different numbers. In the case of proving that 5 divides x^5 - x, you can use the fact that any number raised to the fifth power will be divisible by 5, and subtracting x will not change this divisibility.

3. What is the difference between proving that a number divides another number and proving that it evenly divides it?

There is no difference between these two statements. Both mean that the remainder when dividing the first number by the second number is equal to 0, indicating that the first number is divisible by the second number.

4. Can I use mathematical induction to prove that 5 divides x^5 - x?

Yes, you can use mathematical induction to prove this statement. The base case would be to show that the statement is true for x = 1. Then, assuming that it is true for a particular value of x, you can show that it is also true for x+1. This would prove that 5 divides x^5 - x for all positive integer values of x.

5. Are there any other ways to prove that 5 divides x^5 - x besides using mathematical induction?

Yes, there are other ways to prove this statement. For example, you can use the binomial theorem to expand (x^5 - x) and show that it is divisible by 5. Additionally, you can use modular arithmetic or polynomial division to prove this statement. The method you choose may depend on your level of mathematical knowledge and personal preference.

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