Let x be a positive integer such that 7x + 96 is divisible by x. How many values of x are possible?

1. A. 10
2. B. 11
3. C. 12
4. D. Infinitely many

Answer: C

Explanation

For (7x + 96) to be divisible by x, the expression (7x + 96) / x must result in an integer. We can rewrite this expression as:
(7x / x) + (96 / x) = 7 + (96 / x).
Since 7 is an integer, for the entire expression to be an integer, 96/x must also be an integer. This implies that x must be a divisor (or factor) of 96.
To find the number of positive factors of 96, we first find its prime factorization:
96 = 2 × 48 = 2 × 2 × 24 = 2 × 2 × 2 × 12 = 2 × 2 × 2 × 2 × 6 = 2 × 2 × 2 × 2 × 2 × 3 = 2^5 × 3^1.
The number of factors is calculated by adding 1 to each exponent in the prime factorization and multiplying the results: (5+1) × (1+1) = 6 × 2 = 12.
Thus, there are 12 possible positive integer values for x (1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96).