I4.3 Let $c$ be the remainder of $1^{3}+2^{3}+\ldots+2009^{3}+2010^{3}$ divided by $b^{2}$, find the value of $c$.
Use the formula $1^{3}+2^{3}+\ldots+n^{3}=\frac{1}{4} n^{2}(n+1)^{2}$
$$
\begin{aligned}
1^{3}+2^{3}+\ldots+2009^{3}+2010^{3} & =\frac{1}{4} \cdot 2010^{2} \cdot 2011^{2}=1005^{2} \cdot 2011^{2}=(4 \times 251+1)^{2} \cdot(4 \times 502+3)^{2} \\
& =(4 p+1) \cdot(4 q+1)=4 r+1, \text { where } p, q, r \text { are positive integers. }
\end{aligned}
$$
$\therefore$ When it is divided by $2^{2}$, the remainder is $1, c=1$.