jmc

We know that $3\sqrt{5}=\sqrt{3^2\times 5}=\sqrt{45}$ and $5\sqrt{3}=\sqrt{5^2\times 3}=\sqrt{75}$. There are only two perfect squares between 45 and 75, $7^2=49$ and $8^2=64$, so there are only $\boxed{2}$ integers on the number line between $3\sqrt{5}$ and $5\sqrt{3}$.