Find the number of ordered pairs $(a,b)$ of integers such that $|a + bi| \le 5.$
+129845 a + bi = z
l z l ≤ 5 square both sides
l z l ^2 ≤ 25
a^2 + b^2 ≤ 25
This is a circle centered at (0,0) with a radius of 5
The number of integer coordinates =
1 + 4 (5) + 4 [ floor sqrt (5^2 -1^2) + floor sqrt (5^2 - 2^2) + floor sqrt (5^2 - 3^2) + floor sqrt (5^2 - 4^2) ] =
1 + 20 + 4 [ floor sqrt 24 + floor sqrt (21) + floor sqrt (16) + floor sqrt ( 9) ] =
1 + 20 + 4 [ 4 + 4 + 4 + 3 ] =
1 + 20 + 4 *15 =
1 + 20 + 60 =
81
