Complex Number Real Condition Integer Pairs

For integers $a$ and $b$ consider the complex number $$\frac{\sqrt{ab+2016}}{ab+100}-\left(\frac{\sqrt{|a+b|}}{ab+100}\right)i.$$ Find the number of ordered pairs of integers $(a,b)$ such that this complex number is a real number. [[2016 AIME I Problems/Problem 7 | Solution]]