The maximum volume of a cylinder is cut from a cube of edge $$a$$. The volume of the remaining solid is $$ka^3$$, where $$k = \frac{p}{q}$$, $$\gcd(p, q) = 1$$. Taking $$\pi = \frac{22}{7}$$, the value of $$p + q$$ is

The largest cylinder has radius $$\frac{a}{2}$$ and height $$a$$, so its volume is $$\pi \frac{a^3}{4} = \frac{22}{7} \times \frac{a^3}{4} = \frac{11a^3}{14}$$. The remaining solid has volume $$a^3 - \frac{11a^3}{14} = \frac{3a^3}{14}$$, so $$k = \frac{3}{14}$$ and $$p + q = 3 + 14 = 17$$.