Numbers Reversed when Multiplying by 4

Theorem
Numbers of the form $\sqbrk {21 \paren 9 78}_{10}$ are reversed when they are multiplied by $4$:

and so on.

Proof
Let k represent the number of $9$s in the middle of the number.

For $k > 0$ We can rewrite the number as follows:

Taking numbers of this form and multiplying by $4$ produces:

The first part is composed of $k + 4$ digits. The first two digits will be $84$ followed by $k +2$ digits of $0$

The sum in the middle is composed of $k + 3$ digits. The first digit will be $3$ followed by $k - 1$ digits of $9$ and then the remaining three digits at the end are $600$

Summing the three pieces, the final answer will have $k + 4$ digits.

The first digit is $8$

followed by $7$ which is the sum of the $4$ from the first part and the $3$ of the middle part

followed by $k$ digits of $9$ where the last $9$ is the sum of the $6$ from the middle part and the $3$ of the last part

and then ending in $12$:

Also see

 * Numbers Reversed when Multiplying by 9