The surface area and the volume of a sphere are both 3 digit integers times π. If r is the radius of the sphere, how many integral values can be found for r?

Answer 1

Volume of sphere = (4/3)π r^3
You said the result is a 3 digit integer time π
So r^3 must be an integer if r is to be an integer
perfect cubes are 1,8,27, 64, 125, 216, 343, 512, 729, 1000
4/3 times those must remain a 3 digit integer.
which means the 3 must cancel, implying that the only ones to consider is
(4/3)(216) = 288 and (4/3)(729) = 972

So for the volume part of the question, we may have r = 6 or r = 9

the surface area is 4πr^2
if r = 6, then 4(6)^2 = 144 , ok!
ir r = 9, then 4(9)^2 = 324 , yeah, 3 digits

so r = 6 and r = 9 both satisfy your stated condition.