Question

The difference of a two–digit number and the number obtained by reversing its digits is always divisible by ___________.

Answer 1

Determine the number that divides the difference of a two digits number and its reverse.

A two digits number is expressed as $10x+y$.

So, the number obtained by reversing its digits is $10y+x$.

The difference of the numbers is :

$\begin{array}{rcl}10x+y-\left(10y+x\right)& =& 10x+y-10y-x\\ & \Rightarrow & 10x+y-\left(10y+x\right)=9x-9y\\ & \Rightarrow & 10x+y-\left(10y+x\right)=9\left(x-y\right)\end{array}$

Clearly, the difference is a multiple of $9$.

So, the difference is always divisible by $9$.