Answer
Mass of the core = $5.2\times10^39kg$
It is $2.6\times10^9$ solar masses
Work Step by Step
The speed of an orbiting object is given by $v=\sqrt{\frac{GM}{r}}$. Solve the equation for $M$.
$$v=\sqrt{\frac{GM}{r}}$$
$$M=\frac{rv^2}{G}=\frac{(5.7\times10^17m)(7.8\times10^5m/s)^2}{6.67\times10^{-11}Nm^2/kg^2}=5.2\times10^{39}kg$$
The number of solar masses is found by dividing the mass by the solar mass.
$$=\frac{M_{galaxy}}{M_{Sun}}=\frac{5.2\times10^{39}kg}{2\times10^{30}kg}=2.6\times10^9$$
