Graph, then count (?)

Problem $1$ is called the Gauss Circle Problem . If we have an equation of a circle centered at the origin with radius $R$ , to find the number of integral points within the circle (including the boundary), we have $N(R) = 1 + 4\sum_{i = 0}^{\lfloor{R}\rfloor}\lfloor\sqrt{R^2 - i^2}\rfloor.$ Thus, when $R = 15$ , we get $N(15) = 1 + 4\sum_{i = 0}^{\lfloor{15}\rfloor}\lfloor\sqrt{225 - i^2}\rfloor = 1 + 4(\lfloor{\sqrt{225}}\rfloor + \lfloor{\sqrt{224}}\rfloor + ... + \lfloor{\sqrt{200}}\rfloor + \lfloor{\sqrt{189}}\rfloor +$ $+ ... + \lfloor{\sqrt{176}}\rfloor + \lfloor{\sqrt{161}}\rfloor + \lfloor{\sqrt{144}}\rfloor + \lfloor{\sqrt{125}}\rfloor + \lfloor{\sqrt{104}}\rfloor + \lfloor{\sqrt{81}}\rfloor + \lfloor{\sqrt{56}}\rfloor + \lfloor{\sqrt{29}}\rfloor + 0)$ $= 1 + 4[(15 + 14(5) + 13(2) + 12(2) + 11 + 10 + 9 + 7 + 5] = 1 + 4(177) = 709.$ Remember, $709$ is just the number of integral points within the circle (including the boundary). Hence, we subtract those on the boundary. Finding the number of that particular points is similar to finding the number of integral points $x, y$ satisfying $x^2 + y^2 = 15^2 = 225.$ Clearly, there is only one unordered Pythagorean Triple that satisfies the above equation, that is $(x, y, 15) = (9, 12, 15)$ . But we also need to consider when $x, y = 0.$ Thus, there are a total of $12$ integral points on the boundary, namely $(x, y) = (\pm 12, \pm 9), (\pm 9, \pm 12), (\pm 15, 0), (0, \pm 15).$ The answer in $1$ is then $709 - 12 = 697.$ Now, for $2$ , we are asked to find $697^{1000} \text{ (mod 9)} \equiv 4^{1000} \text{ (mod 9)}.$ Since $\phi(9) = 6$ , we reduce the exponent of $4$ into $1000 \text{ (mod 6)} \equiv 4 \text{ (mod 6)}.$ Thus, that is just simply $4^4 \text{ (mod 9)} \equiv 4 \text{ (mod 9)}.$ With this answer, from $3$ , we are asked to find $S = 1 + \frac{2}{4} + \frac{3}{4^2} + \frac{4}{4^3} + ...$ $\Longrightarrow 4S = 4 + 2 + \frac{3}{4} + \frac{4}{4^2} + ...$ . Subtracting the first equation from the second, we get $3S = 5 + \frac{\frac{1}{4}}{1 - \frac{1}{4}} = \frac{16}{3} \Longrightarrow S = \frac{16}{9} \approx 1.77.$ And so the answer is $\boxed {2}$ .