$a^{2}b=2000=2^{4}\times5^{3}$. Thus, the value of $a$ can be $1$, $2$, $4$, $5$, $10$, and $20$. 
How many solutions that can be expressed with positive integers does the equation below have? ($$1999$$ Math kangaroo Problems, Level $$7-8$$, Question \#$$26$$)  $$a^{2}b-1=1999$$ 
Options:
(1) $$3$$ 
(2) $$4$$ 
(3) $$5$$ 
(4) $$6$$ 
(5) $$7$$ 