Find $abcd$ x 4 = $dcba$

Question

Find $abcd$ x 4 = $dcba$

And in general, find:

$abcd$ x $e$ = $dcba$

This was a problem intended for 3rd graders I believe. I'm having some trouble breaking down an intuitive method that a young student can understand.

Answer 1

First, answer for the specific problem.

$d\geq 4$ because $dcba=4\times abcd\geq 4\times 1000=4000$. Moreover, $a\neq 0$ and $a\leq 2$ because $abcd=\frac{dcba}{4}<2500$ so $d$ is not $4$, $5$, $6$, $7$, $9$. It must be that $d=8$ and $a=2$. So we have $$ (2000+100b+10c+8)\times 4=8000+100c+10b+2\iff 2c=1+13b. $$ It's clear that $b$ must be odd and $1+13b$ is between $0$ and $18$. We conclude that $b=1$ and $c=7$. We have $\boxed{2178\times 4=8712}$.

The general problem can be approached similarly: reason on $d$ and $a$ first (i.e. $d\geq e$ and $a$ is bounded from above because $abcd\times e$ can't exceed $9999$, etc.). Then work on $b$ and $c$ as above.

Answer 2

You can find that divisibility rules might turn out useful in your case.

From Wikipedia

The basic rule for divisibility by 4 is that if the number formed by the last two digits in a number is divisible by 4, the original number is divisible by 4

So, in your case

$4|dcba \implies 4|ba.$