Question

If \( P \) is the greatest divisor of \( 49^n + 16n - 1 \) for all \( n \in \mathbb{N} \), then the number of factors of \( P \) is:

• A. 12
• B. 15
• C. 7
• D. 13

Hint:

To find the number of divisors of a number, express it in prime factorization form and use the formula \( (e_1 + 1)(e_2 + 1) \dots (e_k + 1) \), where \( e_1, e_2, \dots, e_k \) are the exponents of the prime factors.

Answer 1

The Correct Option is C

We are given that \( P \) is the greatest divisor of the expression \( 49^n + 16n - 1 \) for all \( n \in \mathbb{N} \). Our task is to find the number of factors of \( P \).
Step 1: Generalizing the expression
We are tasked with finding the greatest divisor \( P \) of the expression for all \( n \in \mathbb{N} \). The key is to check the values of \( 49^n + 16n - 1 \) for small values of \( n \) and look for any common divisors.
Step 2: Testing for small values of \( n \) - For \( n = 1 \), we compute: \[ 49^1 + 16(1) - 1 = 49 + 16 - 1 = 64 \] - For \( n = 2 \), we compute: \[ 49^2 + 16(2) - 1 = 2401 + 32 - 1 = 2432 \] Next, we find the greatest common divisor (gcd) of 64 and 2432.
Step 3: Finding the gcd of 64 and 2432
- Using the Euclidean algorithm: \[ \gcd(64, 2432) = 64 \] Thus, \( P = 64 \).
Step 4: Finding the number of divisors of \( P \)
The prime factorization of 64 is: \[ 64 = 2^6 \] The number of divisors of \( 64 \) is given by the formula \( (e_1 + 1) \), where \( e_1 \) is the exponent of the prime factor: \[ \text{Number of divisors of } 64 = 6 + 1 = 7 \] Thus, the number of factors of \( P \) is 7.