This solution can be solved on paper if you are a higher dimensional being with amazing calculation skills. So you might need a calculator.
We find the SMALLEST of "An eight-digit integer is formed by repeating a positive four-digit integer.". The reason WHY we find the smallest is because it has the LEAST number of factors, which helps us get to the answer faster.
Which is 10000001
Looking at this, a palindromic integer (i.e. a number same backwards and forwards like 121) with an even amount of digits proves this number is divisible by 11.
Dividing: 10000001 / 11 = 909091
909091 is prime by using the prime checking method.
Now we check if anyother number can be divided by 909091.
25,632,563/909091 = 28.195816480418352
This means taht 909091 cannot be a factor.
Since 11 is the only other factor,
11 is the answer. (probably)