The number of 25^th roots of unity which are also 15^th roots of unity is

from the given condition $e^{i\frac{2k\pi }{25}}=e^{i\frac{2m\pi }{15}}0\le k\le 24,0\le m\le 14$

$\Rightarrow \frac{k}{25}=\frac{m}{15}or3k=5m$

The integral solutions are

K=0,m=0;

k=5,m=3,

k=10,m=6,

k=15,m=9,

and k=20,m=12