How many solutions that can be expressed with positive integers does the equation below have? ($$1999$$ Math kangaroo Problems, Level $$7-8$$, Question \#$$26$$)  $$a^{2}b-1=1999$$ 
Overseas Competition->Knowledge Point->Number Theory Modules->Prime Factorization->Applying Prime Factorization->Finding Factors Given the Product
$a^{2}b=2000=2^{4}\times5^{3}$. Thus, the value of $a$ can be $1$, $2$, $4$, $5$, $10$, and $20$. 