Problem of the Week

Problem 24

Welcome back to Problem of the Week.

For this question, especially part (c), you may find it useful to use the modulo function within your proof. The modulo function, written as “a mod n”, tell you the remainder when a is divided by n. For example, “14 mod 5 = 4”. You may also see it in the form “a $\equiv$ b (mod n)” which means a and b have the same remainder when divided by n.

(a) Show that $x^{3}-x$ is always divisible by 6, where $x$ is a positive integer.

(b) Show that $x^{5}-x$ is always divisible by 30, where $x$ is a positive integer.

(c)* Hence, prove that $x^{p}-x$ is always divisible by $p$ , where $p$ is a prime, and $x$ is a positive integer.

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