A number is said to be a palindrome if reading from right to left is the same as reading from left to right. For example, the numbers 23432 and 18781 are palindromes. How many 4-digit palindrome numbers are divisible by 9?
A 4-digit palindrome number is of the form: $a b b a$, where $a$ is a digit between 1 and 9 and $b$ is a digit between 0 and 9. Since the number is divisible by 9, the sum of its digits: $2a + 2b = 2(a + b)$ is divisible by 9, which means $a + b$ is divisible by 9. If $a + b = 9$, we have the 9 solutions:

$a=1$ and $b=8 \quad ; \quad a=2$ and $b=7 \quad ; \quad a=3$ and $b=6 \quad ; \quad a=4$ and $b=5$

$a=5$ and $b=4 \quad ; \quad a=6$ and $b=3 \quad ; \quad a=7$ and $b=2 \quad ; \quad a=8$ and $b=1$

$a=9$ and $b=0$.

If $a + b = 18$, then the only solution is: $a = b = 9$.

Therefore, the number of 4-digit palindromes divisible by 9 is 10, they are: 1881, 2772, 3663, 4554, 8118, 7227, 6336, 5445, 9009, and 9999.