jmc

If the repeated four-digit integer is $n$, then the eight-digit integer is $10^{4}n+n=10001n$. So all numbers in this form share the factor 10001. Consider $10001\cdot 1000$ and $10001\cdot 1001$. After dividing out the factor 10001, 1000 and 1001 share no nontrivial factors, so the greatest common divisor must be exactly $\boxed{10001}$.