Discussion

Question:

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

Explanation:

The given expression can be written as

$(\frac{16 n^{2}}{n}) + (\frac{7 n}{n}) + (\frac{6}{n}) = 16 n + 7 + (\frac{6}{n})$

Since n is an integer, the expression (16n + 7) will always be an integer. Hence, for the entire expression to be an integer, the part $(\frac{6}{n})$ should also be an integer.
This can be possible only if n is a factor of 6, viz. n = 1, 2, 3, 6, –1, –2, –3 and –6.

Hence, n can have eight values.

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