jmc

The problem asks us to count the number of complex numbers that lie in or on the circle of radius 5 centered at the origin, with integer real and imaginary parts.



We can count that there are 15 such complex numbers in the first quadrant (not including the axes). Then there are 5 complex on the positive real axis, the negative real axis, the positive imaginary axis, and negative imaginary axis. Finally, there is the origin itself, which gives us $4\cdot 15+4\cdot 5+1=\boxed{81}$ complex numbers.