# Find the smallest natural number which is 4 times smaller than the number written with the same digit but in the reverse order.

The question says "Find the smallest natural number which is 4 times smaller than the number written with the same digit but in the reverse order."

I tried to solve it in this way:

New number = 4*(original number)

Thus, new number is a multiple of 4.

SO, its unit diits must be 0,2,4,6,8. Now, how to proceed after this ?

We also know that the first digit of the original number has to be $2$, which means the last digit must be $8$. So we have:
$$8\,a\,b\,2=4\cdot(2\,b\,a\,8)$$
(You can check to see that $3$ digit numbers will not work). It follows that $b$ can only be $0,1,$ or $2$. If we multiply, we obtain the following equation:
$$4a+3\equiv b\operatorname{mod}10\\$$
From this it follows that $a$ is either $2$ or $7$ and $b=1$. This leaves only two options, and one of them works.