Question

For all positive integers \( n \), if \( 3(5^{2n+1}) + 2^{3n+1 \) is divisible by \( k \), then the number of prime numbers less than or equal to \( k \) is:}

• A. \( 17 \)
• B. \( 6 \)
• C. \( 7 \)
• D. \( 8 \)

Hint:

For divisibility problems involving exponents, use modular arithmetic techniques to find the repeating cycle.

Answer 1

The Correct Option is C

Step 1: Find the Divisibility Condition We need to find \( k \) such that: \[ k | \left( 3(5^{2n+1}) + 2^{3n+1} \right) \] By taking the modulo approach and checking divisibility conditions, we find \( k = 17 \). Step 2: Count Prime Numbers \( \leq k \) Prime numbers \( \leq 17 \) are: \[ 2, 3, 5, 7, 11, 13, 17 \] There are 7 prime numbers. Thus, the correct answer is 7.