So Many Constraints!
Number Theory
Level
5
Let \((a,b)\) are positive integers such that
\(\begin{cases} \gcd(a,b)=1, \\ b\le 100, \\ \left(\dfrac{a}{b}-\left[\dfrac{a}{b}\right]\right)\left[\dfrac{a}{b}\right]=2013, \end{cases}\)
where \([x]\) Is the Largest Integer Not Greater Than \(x\)
Find the total number of pairs \((a,b)\).
by
Gabriel Merces