$$Since\ we\ \ are\ looking\ for\ units\ digits,\ we\ only\ need\ to\ evaluate\ 6^{17n}+9^{5m+15n}$$
$$when\ 6\ is\ raised\ to\ the\exp onent\ of\ any\ number,$$
$$the\ unit\ digit\ is\ always\ 6\ so\ unit\ digit\ of\ 6^{17n}\ will\ be\ 6$$
$$we\ are\ left\ with\ 9^{5m+15n}=>\ 9^{5\left(m+3n\right)}$$
$$when\ the\ \exp onent\ of\ 9\ is\ even,\ the\ unit\ digit\ is\ 1\ but\ when\ the\ \exp onent\ of\ 9\ is\ odd,$$
$$the\ unit\ digit\ is\ 9$$
$$so,\ the\ t\arg et\ question\ can\ be\ resrtuctured\ to\ checking\ if\ 5\left(m+3n\right)\ is\ even\ or\ odd$$
$$Statement\ 1\ =>\ 4m+12n=360$$
$$4\left(m+3n\right)=360\ and\ m+3n=90$$
$$m+3n\ is\ even;\ the\ \exp onent\ of\ 9\ is\ 5\left(m+3n\right)$$
$$so,\ it\ is\ 9^{5\left(90\right)}=9^{450}$$
$$\sin ce\ 450\ is\ even,\ the\ unit\ digit\ is\ 1\ so\ the\ answer\ is\ 6\ +\ 1$$
$$statement\ 1\ is\ SUFFICIENT$$
$$Statement\ 2\ =>\ n\ is\ the\ smallest\ 2-digit\ positive\ integer\ divisible\ by\ 5$$
$$the\ smallest\ 2\ digit\ positive\ integer\ divisible\ by\ 5\ =\ 10$$
$$so,\ n=10\ and\ 5\left(m+3n\right)=>5\left[m+3\left(10\right)\right]=>\ 5\left(m+30\right)$$
$$The\ value\ of\ m\ is\ unknown\ so\ the\ \exp onent\ cannot\ be\ evaluated\ to\ decide\ if\ it\ is\ even\ or\ odd$$
$$Statement\ 2\ is\ NOT\ SUFFICIENT$$
$$Since\ statement\ 1\ alone\ is\ SUFFICIENT,$$
$$Answer\ =\ A$$
