Correct Option (C)
The problem states that 7x + 96 is divisible by x, where x is a positive integer. This condition implies that the expression (7x + 96) / x must result in an integer.
The expression can be simplified as follows:
(7x + 96) / x = 7x/x + 96/x = 7 + 96/x
For 7 + 96/x to be an integer, 96/x must also be an integer. This means that x must be a positive divisor (or factor) of 96.
To determine the number of possible values for x, we need to find the total number of positive divisors of 96. First, we obtain the prime factorization of 96:
96 = 2 * 48 = 2 * 2 * 24 = 2 * 2 * 2 * 12 = 2 * 2 * 2 * 2 * 6 = 2^5 * 3^1
The number of positive divisors of an integer is calculated by adding one to each exponent in its prime factorization and then multiplying these results. For 96, the number of divisors is:
(5 + 1) * (1 + 1) = 6 * 2 = 12
Thus, there are 12 possible positive integer values for x that satisfy the given condition.
Incorrect Options:
Options (A) 10 and (B) 11 are incorrect because the systematic calculation of the number of positive divisors of 96 yields 12, not 10 or 11.
Option (D) Infinitely many is incorrect because x must be a divisor of 96. Any positive integer has a finite number of divisors, making an infinite number of possibilities for x impossible in this context.