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 :

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Solution

## 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 asTH H TENS UNIT$\text{y}$ $\text{x}$ $\text{y}$ $\text{x}$ $=1000\text{y+100x+10y+x}$ $=1000\text{y+10y+100x+x}\phantom{\rule{0ex}{0ex}}=1010\text{y+101x}\phantom{\rule{0ex}{0ex}}=101\left(10\text{y+x)}\phantom{\rule{0ex}{0ex}}$So this number is divisible by $101$Hence, a $4-$digit number of this form is divisible by $101$

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