A six digit number is formed by repeating a three digit number for example, 256, 256 or 678, 678 etc. Any number of this form is always exactly divisible by:

The number  $(x  y  z  x  y  z)$ can be written, 
after giving corresponding weightage of the places at which the digits occur,
as $100000 x + 10000y + 1000z + 100x + 10y + z$
$= 100100x + 10010y + 1001z$
$= 1001 (100x + 10y + z)$
Since $1001$ is a factor, the number is divisible by $1001$.
$7 × 11 × 13 = 1001$
As the number is divisible by $1001$, it will also be divisible by all three namely, $7, 11$ and $13$ and not by only one of these because all three are factors of $1001$.

So, the answer is $1001$.

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