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Divisibility is a fundamental concept in mathematics that checks whether one number can be divided by another without leaving a remainder. In the exercise above, we need to prove that \(5^{2n} - 1\) is divisible by 8 for all natural numbers \(n\). This means that when \(5^{2n} - 1\) is divided by 8, it should yield an integer result with no remainder. Understanding divisibility is crucial for solving many mathematical induction problems as it simplifies the expression and proves the desired property. We often check divisibility by performing division and analyzing the remainder. If the remainder is zero, the number is divisible by the divisor. For our example, verifying that expressions like \(8m\) or \(200m + 24\) result in integer values confirms their divisibility by 8.

In mathematical induction, the inductive hypothesis is a crucial assumption. It involves assuming the statement is true for a particular case, often \(n = k\). For the given problem, the inductive hypothesis is that \(5^{2k} - 1\) is divisible by 8, which can be written as \(5^{2k} - 1 = 8m\) for some integer \(m\). The role of the inductive hypothesis is to provide a foundation to prove the next step of induction, known as the inductive step. By assuming the inductive hypothesis holds, we can attempt to prove that the expression is also true when \(n = k + 1\). It is this step that makes mathematical induction a powerful method to prove statements for all natural numbers.

The base case is the starting point of a proof by mathematical induction. It is where you verify that the statement holds true for the initial value, often \(n = 1\). For the exercise, the base case involves checking the expression \(5^{2n} - 1\) for \(n = 1\). Here, the calculation: \(5^{2 \times 1} - 1 = 24\) and checking its divisibility: \(24 \div 8 = 3\), confirms that it is divisible by 8, thus demonstrating the truth of the initial case. The base case is essential since it establishes the foundation upon which the validity of the statement for all subsequent natural numbers depends. If your base case is incorrect, the entire induction proof falls apart.

The inductive step is the process of proving that if the statement holds true for a particular case \(n = k\), it must also be valid for the next case \(n = k + 1\). In our example, we show this by transforming the expression for \(n = k + 1\): \(5^{2(k + 1)} - 1\) simplifies to \(5^{2k + 2} - 1\). By using the inductive hypothesis \(5^{2k} - 1 = 8m\), this expression becomes \(25 \times (8m + 1) - 1 = 200m + 24\). Since both terms \(200m\) and \(24\) are divisible by 8, the entire expression is divisible by 8. Thus, this step validates the indefinite progression of the statement from \(n = k\) to \(n = k + 1\), crucially following from the inductive hypothesis.