[(1)] Let $K$, $L$ be positive odd integers and $A$, $B$ be positive integers satisfying $KA = LB$. Show that if the remainder when $K$ is divided by $4$ equals the remainder when $L$ is divided by $4$, then the remainder when $A$ is divided by $4$ equals the remainder when $B$ is divided by $4$.
[(2)] Let $a$, $b$ be positive integers satisfying $a > b$. Show that there exist positive odd integers $K$, $L$ such that $KA = LB$ holds for $A = {}_{4a+1}C_{4b+1}$, $B = {}_{a}C_{b}$.
[(3)] Let $a$, $b$ be as in (2), and suppose further that $a - b$ is divisible by $2$. Show that the remainder when ${}_{4a+1}C_{4b+1}$ is divided by $4$ equals the remainder when ${}_{a}C_{b}$ is divided by $4$.
[(4)] Find the remainder when ${}_{2021}C_{37}$ is divided by $4$.
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\noindent\textbf{4}
\medskip
\noindent Answer the following questions.
\begin{enumerate}
\item[(1)] Let $K$, $L$ be positive odd integers and $A$, $B$ be positive integers satisfying $KA = LB$. Show that if the remainder when $K$ is divided by $4$ equals the remainder when $L$ is divided by $4$, then the remainder when $A$ is divided by $4$ equals the remainder when $B$ is divided by $4$.
\item[(2)] Let $a$, $b$ be positive integers satisfying $a > b$. Show that there exist positive odd integers $K$, $L$ such that $KA = LB$ holds for $A = {}_{4a+1}C_{4b+1}$, $B = {}_{a}C_{b}$.
\item[(3)] Let $a$, $b$ be as in (2), and suppose further that $a - b$ is divisible by $2$. Show that the remainder when ${}_{4a+1}C_{4b+1}$ is divided by $4$ equals the remainder when ${}_{a}C_{b}$ is divided by $4$.
\item[(4)] Find the remainder when ${}_{2021}C_{37}$ is divided by $4$.
\end{enumerate}
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