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An Introduction to the Theory of Numbers

Exercise 2.1.21 ($n^{12} - 1$ is divisible by $7$ if $(n,7)=1$)

Prove that $n^{12} - 1$ is divisible by $7$ if $(n, 7) = 1$ .

Answers

Proof. Suppose that $n \land 7 = 1$ . By Problem 19, we know that $7 ∣ n^{6} - 1$ .

Since $n^{12} - 1 = (n^{6} - 1) (n^{6} + 1),$

7 ∣ n^{12} - 1 .

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2024-08-21 10:39

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