Number of solutions

Let $I$ be the expression, then we have:

$\begin{aligned} I & = \frac{(2N-6)^3}{N-7} \\ & = \frac{(2[N-7]+8)^3}{N-7} \\ & = \frac{8[N-7] ^3 + 96[N-7]^2 + 384[N-7] + 512}{N-7} \\ & = 8[N-7] ^2 + 96[N-7] + 384 + \color{#3D99F6}{\frac{512}{N-7}} \end{aligned}$

For the LHS be an integer, $\color{#3D99F6}{\frac{512}{N-7}}$ must be an integer or $(N-7)|512$ . Since $512 = 2^9$ ,

$\begin{aligned} \Rightarrow N-7 & = \pm 2^n \quad \quad \small \color{#3D99F6}{\text{where }n = 0,1,2,...9} \\ \Rightarrow N & = \pm 2^n + 7 \end{aligned}$

Therefore, there are $20$ N which give integer $I$ but three of these cases, when $N = 3, 5$ and $6$ , $I \le 0$ (see below), therefore, the number of $N$ that $I$ is a positive integer is $\boxed{17}$ .

$\small \begin{array} {rr} N & I \\ -505 & 2048383 \\ -249 & 500094 \\ -121 & 119164 \\ -57 & 27000 \\ -25 & 5488 \\ -9 & 864 \\ -1 & 64 \\ \color{#D61F06}{3} & \color{#D61F06}{0} \\ \color{#D61F06}{5} & \color{#D61F06}{-32} \\ \color{#D61F06}{6} & \color{#D61F06}{-216} \\ 8 & 1000 \\ 9 & 864 \\ 11 & 1024 \\ 15 & 1728 \\ 23 & 4000 \\ 39 & 11664 \\ 71 & 39304 \\ 135 & 143748 \\ 263 & 549250 \\ 519 & 2146689 \end{array}$