Question

If $x$ is a positive integer such that $2x + 12$ is divisible by $x$, then the number of possible values of $x$ is:

• A. 2
• B. 5
• C. 6
• D. 12

Hint:

Reduce the expression algebraically to isolate the divisibility condition, then find how many integers satisfy it.

Answer 1

The Correct Option is C

We are told: \[ x \mid (2x + 12) \Rightarrow \frac{2x + 12}{x} = \text{integer} \Rightarrow 2 + \frac{12}{x} \in \mathbb{Z} \Rightarrow \frac{12}{x} \in \mathbb{Z} \] So $x$ must divide 12. Positive divisors of 12: \[ x \in \{1, 2, 3, 4, 6, 12\} \Rightarrow \boxed{6} \text{ values} \]