Group Event 8
In the given multiplication, different letters represent different integers whose possible values are $2,4,5,6,7,8$, 9. (Reference: 2000 HI8)
G8.1 Find $A$.
G8.2 Find $B$.
G8.3 Find $C$.
G8.4 Find $D$.
\begin{tabular}{rrrrrrr} 
& 1 & A & B & C & D & E \\
$\times$ & & & & & & 3 \\
\hline & A & B & C & D & E & 1 \\
\hline & 1 & A & B & C & D & 7 \\
$\times$ & & & & & & 3 \\
\hline & A & B & C & D & 7 & 1 \\
\hline & 1 & A & B & C & 5 & 7 \\
$\times$ & & & & & & 3 \\
\hline & A & B & C & 5 & 7 & 1 \\
\hline & 1 & A & B & 8 & 5 & 7 \\
$\times$ & & & & & & 3 \\
\hline & A & B & 8 & 5 & 7 & 1 \\
\hline & 1 & A & 2 & 8 & 5 & 7 \\
$\times$ & & & & & & 3 \\
\hline & A & 2 & 8 & 5 & 7 & 1 \\
\hline & 1 & 4 & 2 & 8 & 5 & 7 \\
$\times$ & & & & & & 3 \\
\hline & 4 & 2 & 8 & 5 & 7 & 1
\end{tabular}
$\mathrm{E}=7$ and the carry digit is 20 .
$3 \mathrm{D}+2 \equiv 7(\bmod 10)$
$\mathrm{D}=5$ and the carry digit is 10
$3 \mathrm{C}+1 \equiv 5(\bmod 10)$
$\mathrm{C}=8$ and the carry digit is 20 .
$3 \mathrm{~B}+2 \equiv 8(\bmod 10)$
$\mathrm{B}=2$ and there is no carry digit.
$3 \mathrm{~A} \equiv 2(\bmod 10)$
$\mathrm{A}=4$ and the carry digit is 10 .
$3+1=4$
$\therefore A=4, B=2, C=8, D=5$