Question

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

• A. 2
• B. 3
• C. 4
• D. None of these

Hint:

For rational expressions, integrality requires denominator to divide remainder in division.

Answer 1

The Correct Option is C

$\frac{16n^{2} + 7n + 6}{n} = 16n + 7 + \frac{6}{n}$. For this to be integer, $\frac{6}{n}$ must be integer $\Rightarrow n$ divides 6. Possible integer divisors: $\pm 1, \pm 2, \pm 3, \pm 6$ $\Rightarrow$ 8 values. But if $n=0$, expression undefined, so exclude. All divisors give integer value, so answer = 8, not in Option → correct is "None of these".