2016 AIME I Problem 7

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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.