Question

If \(\frac{5^4 - 1}{n}\) is an integer an n is an integer, then n could be each of the following EXCEPT

• A. 4
• B. 6
• C. 13
• D. 25
• E. 26

Hint:

For "EXCEPT" questions, the process is one of elimination. Test each option against the condition. Using algebraic identities like the difference of squares can make simplifying the initial expression much faster than direct calculation.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The problem states that \(\frac{5^4 - 1}{n}\) is an integer. This means that \(n\) must be a factor (or divisor) of the numerator, \(5^4 - 1\). The question asks which of the given options is NOT a factor of \(5^4 - 1\).
Step 2: Key Formula or Approach:
We will first calculate the value of the numerator. A useful algebraic identity is the difference of squares: \(a^2 - b^2 = (a-b)(a+b)\).
Step 3: Detailed Explanation:
First, let's simplify the numerator, \(5^4 - 1\).
We can write this as \((5^2)^2 - 1^2\).
Using the difference of squares formula:
\[ (5^2)^2 - 1^2 = (5^2 - 1)(5^2 + 1) \]
Now, calculate the terms in the parentheses:
\[ (25 - 1)(25 + 1) = (24)(26) \]
So, the numerator is \(24 \times 26 = 624\).
The problem now is to determine which of the options is not a factor of 624. Let's test each option.
\begin{itemize} \item (A) 4: Since \(24 = 4 \times 6\), 4 is a factor of 24, and therefore a factor of 624. (\(624 \div 4 = 156\)). \item (B) 6: Since \(24 = 6 \times 4\), 6 is a factor of 24, and therefore a factor of 624. (\(624 \div 6 = 104\)). \item (C) 13: Since \(26 = 13 \times 2\), 13 is a factor of 26, and therefore a factor of 624. (\(624 \div 13 = 48\)). \item (D) 25: To be divisible by 25, a number must end in 00, 25, 50, or 75. The number 624 ends in 24, so it is not divisible by 25. \item (E) 26: From our calculation, \(624 = 24 \times 26\), so 26 is clearly a factor. (\(624 \div 26 = 24\)). \end{itemize} Step 4: Final Answer:
The number 25 is not a factor of 624, so n cannot be 25.