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Math Help - [SOLVED] What is the 4-digit number?

  1. #1
    MSYORK89
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    Unhappy [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?
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  2. #2
    Super Member

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    Hello, MSYORK89!

    Is there a typo?


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

    . . \begin{array}{cccc}<br />
1 & 2 & 3 & 4 \\ \hline<br />
A & B & C & D \\<br />
\times & & & A \\ \hline<br />
D & C & B & A<br />
\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}<br />
1 & 2 & 3 & 4 \\ \hline<br />
2 & B & C & D \\<br />
\times & & & 2 \\ \hline<br />
D & C & B & 2<br />
\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}<br />
1 & 2 & 3 & 4 \\ \hline<br />
3 & B & C & D \\<br />
\times & & & 3 \\ \hline<br />
D & C & B & 3<br />
\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.


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


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

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

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