Question

A $4-$digit number is of the formed by repeating a $2-$digit number such as $2525,3232$ etc. Any number of the form is exactly divisible by :

Answer 1

Formation of the number :

Let the unit digit be $\text{x}$ and the tens digit be $\text{y}$

So a $4-$digit number is of the form $2525$ can be written as

TH H TENS UNIT

$\text{y}$ $\text{x}$ $\text{y}$ $\text{x}$ $=1000\text{y+100x+10y+x}$

$$\begin{gathered} =1000\text{y+10y+100x+x} \\ =1010\text{y+101x} \\ =101(10\text{y+x)} \end{gathered}$$

So this number is divisible by $101$

Hence, a $4-$digit number of this form is divisible by $101$