A four-digit number is made by writing the same two-digit number twice over, as in 5757 or 2323. Every number built this way is exactly divisible by:

If the two-digit block is $10a+b$, the four-digit number is $1000a+100b+10a+b=1010a+101b=101\times (10a+b)$. So it is always 101 times the block — for instance $5757=101\times 57$. The tempting answer 11 works only sometimes: 5757 ÷ 11 is not a whole number, though 2323 ÷ 11 = 211 happens to be. The six-digit version (abcabc) is the one divisible by 1001.