Divisibility Rules

We are given that is the greatest divisor of the expression for all . Our task is to find the number of factors of .
Step 1: Generalizing the expression
We are tasked with finding the greatest divisor of the expression for all . The key is to check the values of for small values of and look for any common divisors.
Step 2: Testing for small values of - For , we compute: - For , we compute: 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: Thus, .
Step 4: Finding the number of divisors of
The prime factorization of 64 is: The number of divisors of is given by the formula , where is the exponent of the prime factor: Thus, the number of factors of is 7.