Tim from Gravesend Grammar School and Mohammad Afzaal Butt both sent us similar solutions to the problem. Well done Tim and Mohammad. Here is Mohammad's solution:

Let the three digit number be $xyz$. Hence the six digit number will be $xyzxyz$. Now
$$\eqalign { xyzxyz  &= 100000x + 10000y + 1000z + 100x + 10y + z \cr
&= 100100x + 10010y + 1001z \cr
&= 1001 (100x + 10y + z) \cr
&= 7 \times 11 \times 13 (100x + 10y + z)} $$ Hence the number $xyzxyz$ is always divisible by $7$, $11$ and $13$.