Question

A six-digit number is formed by repeating a three-digit number; for example, $256256$ or $678678$, etc. any number of this form is always exactly divisible by :

Answer 1

$\Rightarrow$From the given question, the number should be of the form $\mathrm{abcabc}$

$\therefore$ The general form of $\mathrm{abcabc}$ $=100000\mathrm{a}+10000\mathrm{b}+1000\mathrm{c}+100\mathrm{a}+10\mathrm{b}+\mathrm{c}$

By simplifying the above expression, we will get:

$\begin{array}{rcl}& \Rightarrow & \mathrm{abcabc}=100100\mathrm{a}+10010\mathrm{b}+1001\mathrm{c}\\ & =& 1001\left(100\mathrm{a}+10\mathrm{b}+\mathrm{c}\right)\end{array}$

$=1001\times \mathrm{n},\mathrm{where}\mathrm{n}\mathrm{is}\mathrm{a}\mathrm{natural}\mathrm{number}\mathrm{which}\mathrm{depends}\mathrm{on}\mathrm{the}\mathrm{values}\mathrm{of}\mathrm{a},\mathrm{b}\mathrm{and}\mathrm{c}$ ($\because 100\mathrm{a}+10\mathrm{b}+\mathrm{c}=\mathrm{n}$)

Therefore, the number will be exactly divisible by $1001$