For integers a and b consider the complex number

\frac{\sqrt{a b+2016}}{a b+100}-\left(\frac{\sqrt{|a+b|}}{a b+100}\right) i

Find the number of ordered pairs of integers (a, b) such that this complex number is a real number.

For integers a and b consider the complex number

\frac{\sqrt{a b+2016}}{a b+100}-\left(\frac{\sqrt{|a+b|}}{a b+100}\right) i

Find the number of ordered pairs of integers (a, b) such that this complex number is a real number.