<s>[INST] Please help me to find the value of $ABCDE$ (step by step) $ABCD\times E = DCBA$ with $A,B,C,D$, and $E$ distinct decimal digits (and $ABCD$ representing the concatenation of those digits).  How can I find the value each of them?
 [/INST] 2178*4 = 8712 
Consider $E$ cannot be 1. otherwise $D=A$.If $E\ge 5$, then $A=1$, $D=E$, contradiction!
So $E\ge 2$, thus we have 


*

*$E = 2$, then $A$ can only be $1,2,3,4$, since the product should be
4-digit number.
$DE$ has end digit as $A$, then $A$ is even, $A$ can be 2,4, since $A\neq E$, therefore $A=4$. Then $D = 7$($D\neq E$), but we look at the first digit of $\overline{DCBA}$, we should know that $D\ge E*A = 8$, contradiction!

*$E=3$, then $A$ can be $1,2,3$, and we exclude $1,3$ for the same reason. $A=2$, thus $D = 4$, But we expect $D\ge E*A = 6$.

*$E=4$, thus $A = 2$, $DE$ has end digit as $2$, thus $D=8$ or $D=3$, and $D\ge E*A =8$, THUS $D=8$.


Then we solve the other two unknowns by solving an equation:
$8000+400B+40C+32 = 8000+100C+10B+2 $,
which is
$39B+3 = 6C$, where we can see $B\le 1$, since $C\le 9$.
we have $(B,C) = (1,7)$, ONLY.
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