A palindrome is a positive integer that reads the same forward and backward, like 2552 or 1991. Find a positive integer greater than 1 that divides all four-digit palindromes.
We claim that 11 divides all four digit palindromes. Note that any four digit palindrome $a b b a$ is the sum

$$
a b b a=a 00 a+b b 0=a \times 1001+b \times 110 .
$$

Now, 110 is a multiple of 11 , since it is $11 \times 10$, and 1001 is a multiple of 11 , since it is $11 \times 91$. Thus, $a 00 a$ and $b b 0$ are multiples of 11 , so their sum is also a multiple of 11 .