7 divides 23n-1

Let the given statement be P(n), i.e.,

P(n): 23n – 1 is divisible by 7.

It can be observed that P(n) is true for n = 1 since 23 × 1 –  1 = 8 – 1 = 7, which is divisible by 7.

Let P(k) be true for some positive integer k, i.e.,

P(k): 23k – 1 is divisible by 7.

∴23k – 1 = 7m; where m ∈ N … (1)

We shall now prove that P(k + 1) is true whenever P(k) is true.

Consider

Therefore, 23n – 1 is divisible by 7.

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

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