$x$ is a positive integer such that $x^2 + 12$ is exactly divisible by $x$. Find all the possible values of $x$.

### Concept & Logic When a variable divides a polynomial, split the polynomial into individual fractions. For the final expression to be an integer, every resulting fraction must simplify to an integer. ### Step-by-Step Solution - **Calculation / Deduction:** Split the given algebraic expression into two separate fractions: $$ \frac{x^2 + 12}{x} = \frac{x^2}{x} + \frac{12}{x} = x + \frac{12}{x} $$ - Since $x$ is a positive integer, the first term $x$ is always an integer. - For the entire expression to be an integer, the second term $\frac{12}{x}$ must also result in a whole number. - This means $x$ must be a factor of 12. - The positive factors of 12 are 1, 2, 3, 4, 6, and 12. ### Exam Strategy & Shortcut Whenever you see a pattern like $\frac{x^2 + k}{x}$, instantly ignore the $x^2$ term. The possible integer values for $x$ will always simply be the positive factors of the constant $k$. ### Common Pitfall Students often forget that 1 and the number itself (12) are valid factors. Always list factor pairs systematically ($1 \times 12$, $2 \times 6$, $3 \times 4$) to avoid missing the boundary values. ### Final Answer Therefore, the correct answer is 1, 2, 3, 4, 6, 12.