# [SOLVED] What is the 4-digit number?

• Oct 31st 2007, 02:45 PM
MSYORK89
[SOLVED] What is the 4-digit number?
If ABCD x A= DCBA, what are the four digit numbers that give you this? And what is the equation? Without repitition?
• Nov 1st 2007, 06:36 PM
Soroban
Hello, MSYORK89!

Is there a typo?

Quote:

If $ABCD \times A\:=\: DCBA$, what are the four-digit numbers?
We have:

. . $\begin{array}{cccc}
1 & 2 & 3 & 4 \\ \hline
A & B & C & D \\
\times & & & A \\ \hline
D & C & B & A
\end{array}$

If $A =0$, the product is zero. .Hence, $A \neq 0.$

If $A = 1$, the product is $ABCD.$ .Hence, $A \neq 1.$

If $A \geq 4$, the product is a five-digit number. .Hence, $A \:=\:2\text{ or }3$

If $A = 2$, we have:

. . $\begin{array}{cccc}
1 & 2 & 3 & 4 \\ \hline
2 & B & C & D \\
\times & & & 2 \\ \hline
D & C & B & 2
\end{array}$

In column-4, $D \times 2$ ends in 2.
. . Hence, $D \,=\,1\text{ or }6$

But in column-1, we see that $D\,=\,4\text{ or }5$

. . Therefore: . $A \,\neq\,2$

If $A = 3$, we have:

. . $\begin{array}{cccc}
1 & 2 & 3 & 4 \\ \hline
3 & B & C & D \\
\times & & & 3 \\ \hline
D & C & B & 3
\end{array}$

In column-4, we see that: . $D = 1$

But in column-1, $D \,=\,9$

. . Therefore: . $A \,\neq\,3$

The problem has no solutions.

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If the problem is: . $ABCD \times {\color{red}4} \:=\:DCBA$

. . there is a solution: . $2178 \times 4 \:=\:8712$