Understanding the Divisibility of \(5^{2n} - 1\)

The question asks us to find a number that always divides the expression \(5^{2n} - 1\) for any positive integer value of \(n\).

We can analyze the expression \(5^{2n} - 1\) in a couple of ways: using algebraic factorization and by testing small values of \(n\).

Method 1: Algebraic Factorization of \(5^{2n} - 1\)

The expression can be rewritten using exponent rules:

\[ 5^{2n} - 1 = (5^2)^n - 1 \] \[ = 25^n - 1 \]

Now, we can use the algebraic identity for the difference of powers, which states that \(a^n - b^n\) is always divisible by \(a - b\) for any positive integer \(n\). In this case, we have \(a = 25\) and \(b = 1\).

Applying the identity, \(25^n - 1^n\) is divisible by \(25 - 1\).

\[ 25 - 1 = 24 \]

So, \(25^n - 1\) is always divisible by 24. The expression \(25^n - 1\) can be factored as:

\[ 25^n - 1^n = (25 - 1)(25^{n-1} + 25^{n-2} \cdot 1 + \dots + 25 \cdot 1^{n-2} + 1^{n-1}) \] \[ = 24 (25^{n-1} + 25^{n-2} + \dots + 25 + 1) \]

Since \(n\) is a positive integer, the second factor \((25^{n-1} + 25^{n-2} + \dots + 25 + 1)\) is a sum of positive integers (or just 1 when \(n=1\)), which results in an integer. Therefore, the expression \(5^{2n} - 1\) is always equal to 24 multiplied by an integer, meaning it is always divisible by 24.

Method 2: Testing Small Values of \(n\) for Divisibility

Let's check the expression for the first few positive integer values of \(n\):

• For \(n = 1\): \(5^{2 \cdot 1} - 1 = 5^2 - 1 = 25 - 1 = 24\).
• For \(n = 2\): \(5^{2 \cdot 2} - 1 = 5^4 - 1 = 625 - 1 = 624\).
• For \(n = 3\): \(5^{2 \cdot 3} - 1 = 5^6 - 1 = 15625 - 1 = 15624\).

Now let's check if these values are divisible by the given options:

• Option 1: 25
  • 24 is not divisible by 25.
• Option 2: 15
  • 24 is not divisible by 15.
• Option 3: 24
  • 24 is divisible by 24 (\(24 \div 24 = 1\)).
  • 624 is divisible by 24 (\(624 \div 24 = 26\)).
  • 15624 is divisible by 24 (\(15624 \div 24 = 651\)).
• Option 4: 42
  • 24 is not divisible by 42.

From testing, it is clear that 24 divides the expression for \(n=1, 2, 3\), while the other options do not divide the expression for \(n=1\).

Both methods confirm that the expression \(5^{2n} - 1\) is always divisible by 24 for any positive integer \(n\).