If a circle is centered on the origin and has radius r, it may pass through some points with integer coordinates. In fact, by symmetry, this will be a multiple of 4.
Find the smallest radius that will pass through 4n integer points where n=1,2,3,4,5.
Feel free to go further.
(In reply to re: n = 2
by Ady TZIDON)
Ah yes, now that I understand, I do agree with your solution for n = 1,2,3,4. Full marks!