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An Introduction to the Theory of Numbers

Exercise 2.1.22 ($n^{6k} - 1$ is divisible by $7$ if $(n,7) = 1$)

Prove that $n^{6 k} - 1$ is divisible by $7$ if $(n, 7) = 1$ , $k$ being any positive integer.

Answers

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

Since

n^{6 k} - 1 = (n^{6} - 1) \sum_{i = 0}^{k - 1} n^{6 i},

for any positive integer $k$ ,

7 ∣ n^{6 k} - 1 .

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2024-08-21 10:41

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