Would dividing a random number by a constant "ruin" it? If I get a random number from a cryptographically secure random number generator then divide it by a number, say 2, is it still cryptographically secure?
It does not lower the security at all - at most, it changes the output space. If your output space is, say, the integers between $-100$ and $100$, then computing PRG$(s)/2$ for a random seed $s$ gives a value indistinguishable from a random value in the set $\{-50, -49.5, \cdots, 49.5,50\}$. Depending on which output space you consider and what you exactly mean by "dividing by 2", it can even remain a cryptographically strong PRG with the same output space: if your output space is, say, the field $\mathbb{F}_p$ with $p$ prime, then computing PRG$(s)$ and multiplying the output by $2^{-1} \bmod p$ gives a value indistinguishable from random in $\mathbb{F}_p$.

This follows immediatly from the security of the original PRG: since division by 2 is a reversible operation, any distinguishing attack on the new PRG trivially translates into a distinguishing attack against the original PRG.