jmc

The product of the (positive and negative) divisors of an integer $a$ is negative if $a$ has an odd number of negative divisors. It follows that $-a$ must have an odd number of positive divisors. However, for every positive divisor $d$ of $-a$, then $(-a)/d$ is also a positive divisor of $-a$, so that the positive divisors of $-a$ can be paired up. The exception is if $-a$ is a perfect square, in which case $\sqrt{-a}$ will not be paired up with another divisor. There are $\boxed{14}$ perfect squares between $1$ and $200$: $1^2,2^2,3^2,\cdots ,14^2=196$.