The number of common tangents that can be drawn to the circles $x^2+y^2=1$ and $x^2+y^2-2x-6y+6=0$ is 

For the circle $x^2+y^2=1 ;$ Centre $=C_1(0,0),\text{ radius }{r}_1=1.$

For the circle $x^2+y^2-2x-6y+6=0;$ Centre $C_2=(1,3),$ radius $r_2=\sqrt{1+9-6}=2.$

$C_1{C}_2=\sqrt{1+9}=\sqrt{10}>r_1+r_2.\therefore one circle completely lies out side$ 

The number of common tangents $=4.$