Can you please provide hints? A 6 digit number can be written by repeating a three digit number, such as 359, 359. What is the greatest integer which divides any such 6 digit number? I can see that the number is not guaranteed to be divisible by 2 or by 3, so the answer will not be a multiple of 2, 3, or 6. However, I do not see where to go from here (by the way please do not use modular arithmetic in your answer).
Let $x$ be some three digit number. We have:
$$ 1000x + x $$ $$ 1001x $$
Now you know that $x$ may be any number, so it doesn't necessarily have any other common factors. (Consider any two prime numbers.) Our answer is $1001$.
It is clear that $1,001$ divides all such numbers, since $abc,abc=abc\cdot 1,001.$ Note then that $101$ and $103$ are prime, so the largest common factor of $101,101$ and $103,103$ is $1,001$. Thus, the largest integer that divides all such numbers is $1,001$.