4x(ABCD)=DCBA what is ABCD ?

Nov 10, 2017

$A B C D = 2178$

Explanation:

If $A$, $B$, $C$, $D$ are digits of a $4$ digit number, then we can reason as follows:

Given:

$4 A B C D = D C B A$

• $A$ is $1$ or $2$, since otherwise $4 A B C D$ would run to $5$ digits.

• $A$ is even since it's the last digit of $4 A B C D$, so that means it must be $2$.

• $D \ge 8$ and hence $D = 8$ or $9$.

• Since the last digit of $4 D$ is $A$, we can deduce that $D = 8$.

• So there are no carries into the $1000$'s and there is a carry of $3$ into the $10$'s. So $4 B C + 3 = C B$

• Hence $B$ is odd and $1$ or $2$, so must be $1$.

• Then the last digit of $4 B C + 3$ is $1$, so the last digit of $4 B C$ is $8$ and $C = 2$ or $C = 7$. Hence $C = 7$

$A B C D = 2178$

Nov 10, 2017

If $A B C D$ AND $D C B A$ were intended to be 4-digit numbers (with $A , B , C , \mathmr{and} D$ representing single digits:
$\textcolor{w h i t e}{\text{XXX}} A B C D = 2178$

Explanation:

$4 \times 2178 = 8712$

(only non-zero combination that I was able to find)

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