Final answer:
We will prove that (23)ⁿ - 1 is divisible by 11 for all positive integer n using mathematical induction.
Step-by-step explanation:
We will prove that (23)ⁿ - 1 is divisible by 11 for all positive integer n using mathematical induction.
Base Case: For n = 1, we have (23)¹ - 1 = 23 - 1 = 22, which is divisible by 11.
Inductive Hypothesis: Assume the statement holds for some positive integer k, i.e., (23)ᵏ - 1 is divisible by 11.
Inductive Step: Now, we need to prove the statement holds for k + 1.
To do this, let's consider (23)^(k+1) - 1:
(23)^(k+1) - 1 = (23)ᵏ * 23 - 1 = (23)ᵏ * 22 + (23)ᵏ * 1 - 1 = 22 * (23)ᵏ + (23)ᵏ - 1
Since (23)ᵏ - 1 is divisible by 11 (by the inductive hypothesis), we can write it as (23)ᵏ - 1 = 11m, for some integer m.
Substituting into the above equation, we get (23)^(k+1) - 1 = 22 * (11m) + (23)ᵏ - 1 = 11(2m * 22 + (23)ᵏ - 1).
Therefore, (23)^(k+1) - 1 is divisible by 11 for the case k + 1.
By the principle of mathematical induction, we have proved that (23)ⁿ - 1 is divisible by 11 for all positive integer n.