4x(ABCD)=DCBA what is ABCD ?

Question

13089 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-11-10T13:23:58

$ABCD=2178$

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

Given:

$4ABCD = DCBA$

$A$ is $1$ or $2$ , since otherwise $4ABCD$ would run to $5$ digits.
$A$ is even since it's the last digit of $4ABCD$ , so that means it must be $2$ .
$D >= 8$ and hence $D=8$ or $9$ .
Since the last digit of $4D$ 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 $4BC+3 = CB$
Hence $B$ is odd and $1$ or $2$ , so must be $1$ .
Then the last digit of $4BC+3$ is $1$ , so the last digit of $4BC$ is $8$ and $C=2$ or $C=7$ . Hence $C=7$

$ABCD=2178$

Answer 2 · 2017-11-10T13:41:08

If $ABCD$ AND $DCBA$ were intended to be 4-digit numbers (with $A, B, C, and D$ representing single digits:
$color(white)("XXX")ABCD=2178$

$4xx2178=8712$

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

Is this what you were trying to ask for?