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

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


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

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

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


    If $\displaystyle A = 2$, we have:

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

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

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

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


    If $\displaystyle A = 3$, we have:

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

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

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

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


    The problem has no solutions.


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


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

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

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