Question

A six-digit number is formed by repeating a three-digit number. For example$256256$, $678678$etc. Any number of this form is divisible by:

• A. $7$ only
• B. $11$ only
• C. $13$ only
• D. $1001$

Answer 1

The correct option is D

$1001$

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:

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

$=1001\times \mathrm{n}$ where $\mathrm{n}$ is a natural number whose value depends upon values of $\mathrm{a},\mathrm{b},\mathrm{c}$ ($\because 100\mathrm{a}+10\mathrm{b}+\mathrm{c}=\mathrm{n}$)

So, the number should be divisible by $1001$

Therefore, the correct option is D.