Let x be a positive integer such that 7x + 96 is divisible by x . How many values of x are possible?
- A10
- B11
- C12Correct
- DInfinitely many
Explanation
To determine the number of possible values for x, we look at the expression (7x + 96) / x.
This can be split into two parts: (7x / x) + (96 / x).
Simplifying this gives: 7 + (96 / x).
For this expression to result in an integer, 96 must be divisible by x. Therefore, x must be a factor of 96.
To find the number of factors of 96, we first perform prime factorization: 96 = 2^5 * 3^1.
The total number of factors is calculated by adding 1 to each exponent and multiplying them: (5 + 1) * (1 + 1) = 6 * 2 = 12.
Since there are 12 factors of 96, there are 12 possible values for the positive integer x.
The correct answer is C.
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