For how many integers $N$ between $1$ and $1990$ is the improper fraction $\frac{N^2+7}{N+4}$ $\underline{not}$ in lowest terms?
$\text{(A) } 0\quad \text{(B) } 86\quad \text{(C) } 90\quad \text{(D) } 104\quad \text{(E) } 105$
What we want to know is for how many $n$ is \[\gcd(n^2+7, n+4) > 1.\] We start by setting \[n+4 \equiv 0 \mod m\] for some arbitrary $m$. This shows that $m$ evenly divides $n+4$. Next we want to see under which conditions $m$ also divides $n^2 + 7$.  We know from the previous statement that \[n \equiv -4 \mod m\] and thus \[n^2 \equiv (-4)^2 \equiv 16 \mod m.\] Next we simply add $7$ to get \[n^2 + 7 \equiv 23 \mod m.\] However, we also want \[n^2 + 7 \equiv 0 \mod m\] which leads to \[n^2 + 7\equiv 23 \equiv 0 \mod m\] from the previous statement. From that statement, we get that $m$ divides $23$ evenly. Since $23$ is prime and we're looking for a GCD greater than 1,  $m$ must be $23$. Going back to our original statement, we can set \[n+4=23x\] for some positive integer x, and \[n=23x-4.\] Finally, we must find the largest $x$ such that \[23x-4<1990.\] This is a simple linear inequality for which the answer is $x=86$, or $\fbox{B}$.