a/b game

Solution :

$a^2-2a<b<a^2 \rightarrow -2a<b-a^2<0 \rightarrow b=ak \rightarrow -2a<ak-a^2<0$

$\rightarrow -2a<a(k-a)<0 \rightarrow -2<k-a<0 ; k-a \in \mathbb{Z} \rightarrow k-a=-1$

$ab<10000 \rightarrow a^2 (a-1) <10000 \rightarrow \text {by a little trial and error}$

$\rightarrow a \leq 21$

$1<ab \rightarrow 1<a^2 (a-1) \rightarrow 2 \leq a \rightarrow 2 \leq a \leq 21$

$\rightarrow \text{there are a total of 20 instances}$