Question

If n is a natural number less than or equal to 5, for how many values of n is 3ⁿ- n a prime number?

• A. 0
• B. 1
• C. 2
• D. 3
• E. 4

Answer 1

The Correct Option is D

To solve the problem, we need to determine for which values of \(n\) (where \(n\) is a natural number and \(n \leq 5\)), the expression \(3^n - n\) results in a prime number. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

1. Calculate \(3^n - n\) for each value of \(n\) from 1 to 5:
   • For \(n = 1\):
     \(3^1 - 1 = 3 - 1 = 2\)
     2 is a prime number.
   • For \(n = 2\):
     \(3^2 - 2 = 9 - 2 = 7\)
     7 is a prime number. 
   • For \(n = 3\):
     \(3^3 - 3 = 27 - 3 = 24\)
     24 is not a prime number (it has divisors other than 1 and itself).
   • For \(n = 4\):
     \(3^4 - 4 = 81 - 4 = 77\)
     77 is not a prime number (it can be divided by 7 and 11).
   • For \(n = 5\):
     \(3^5 - 5 = 243 - 5 = 238\)
     238 is not a prime number (it can be divided by 2).
2. We found that for \(n = 1\) and \(n = 2\), \(3^n - n\) is a prime number. Therefore, the expression is prime for only two values of \(n\).

Therefore, for the given range of natural numbers, \(3^n - n\) is a prime number for 2 values of \(n\). Thus, the correct answer is:

2