5. A four-digit number has the following property: dividing this four-digit number by its last two digits yields a perfect square (if the tens digit is zero, then divide by the units digit), and this perfect square is exactly the square of the first two digits plus 1. For example, $4802 \div 2$ $=2401=49^{2}=(48+1)^{2}$. Then the smallest four-digit number with the above property is $\qquad$
5. 1085 Let this four-digit number be $100 c_{1}+c_{2}$, where $10 \leqslant c_{1} \leqslant 99, 1 \leqslant c_{2} \leqslant 99$. According to the problem, we have
$$
\begin{array}{l} 
100 c_{1}+c_{2}=\left(c_{1}+1\right)^{2} c_{2}=c_{1}^{2} c_{2}+2 c_{1} c_{2}+c_{2}, \\
\therefore 100 c_{1}=c_{1} c_{2}\left(c_{2}+2\right),
\end{array}
$$

Therefore, $\left(c_{1}+2\right) \mid 100$, and $10 \leqslant c_{1} \leqslant 99$,
then $c_{1}=\{18,23,48,98\}, c_{2}=\{5,4,2,1\}$.
Thus, the four-digit numbers are $1805, 2504, 4802, 280 \%$, and the smallest four-digit number among them is 1805.