If $n$ is an integer, how many values of $n$ will give an integral value of $\frac{16n^2 + 7n + 6}{n}$?

### Concept & Strategy To determine when a rational expression yields an integer, divide each term in the numerator by the denominator to separate the polynomial part from the strictly fractional part. $$\frac{an^2 + bn + c}{n} = an + b + \frac{c}{n}$$ ### Step-by-Step Solution 1. Given expression: $\frac{16n^2 + 7n + 6}{n}$. 2. Separate the terms: $\frac{16n^2}{n} + \frac{7n}{n} + \frac{6}{n}$. 3. Simplify: $16n + 7 + \frac{6}{n}$. 4. Since $n$ is an integer, $16n$ and $7$ are always integers. 5. For the entire expression to be an integer, the remaining term $\frac{6}{n}$ must be an integer. 6. This means $n$ must be a factor (divisor) of 6. 7. The integer divisors of 6 are $1, 2, 3, 6$ and their negative counterparts $-1, -2, -3, -6$. 8. Count the values: There are exactly 8 values for $n$. ### Exam Strategy & Shortcut Mentally split the fraction. Ignore the terms with $n$ in the numerator as they will perfectly divide. Focus purely on the constant term divided by $n$ (here, $\frac{6}{n}$). Count the total divisors (both positive and negative) of the constant term. Total divisors of 6 = $4 \times 2 = 8$. ### Common Pitfall Forgetting to include negative integers. A student might only count the positive divisors (1, 2, 3, 6) and erroneously look for an option representing 4, completely missing half the valid solutions. ### Final Answer **Therefore, the correct answer is None of these.**