# I'm invoking the Fractional-Part Function!

**Number Theory**Level 4

\[\large{ \{q\}^2 = \left \{ \dfrac{n!}{2000} \right \} }\]

How many pairs \((n,q)\) satisfy the above equation, where \(n\) is a positive integer and \(q\) is a non-integer rational number such that \( 0<q<2000\)?

**Notation**: \(\{r\}\) denotes the fractional part function.