The first circle passes through (2, 0) and (0, 2), and since it passes through the origin, its equation is x squared plus y squared minus 2x minus 2y equals 0, making its center (1, 1) with a radius of root 2. The second circle has the segment from the origin to (1, 1) as its diameter, so its center is (0.5, 0.5) and its radius is root 2 divided by 2. The distance between their centers is root 2, which equals the difference of their radii, meaning the circles touch internally. Two circles that touch internally have exactly one common tangent in the plane. Therefore, the number of common tangents is 1.