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Algebra Difficulty 4.7 AIME Prove it Saudi Arabia

Let $M$ and $N$ are two nine-digit palindromes without leading zeros (a palindrome is a number that reads the same backward as forward) such that $M < N$ and there are no other nine-digit palindromes strictly between $M$ and $N$ . What are all possible values of $N - M$ ?

Solution

Denote the number $M$ is $a b c d e d c b a$ . We consider cases:

1. If $e \geq 8$ then the next number $N$ is $a b c d (e+1) d c b a$ .

2. If $e = 9$ , $d \geq 8$ then the next number $N$ is $a b c (d+1) 9 (d+1) c b a$ .

3. If $e = d = 9$ , $c \leq 8$ then the next number $N$ is $a b (c+1) 999 (c+1) b a$ .

4. If $e = d = c = 9$ , $b \leq 8$ then next number is $a (b+1) 99999 (b+1) a$ .

5. If $e = d = c = b = 9$ , then $a \leq 8$ (otherwise, there is no $N$ ). Then the next number is $(a+1) 9999999 (a+1)$ .

So the difference are $10000$ , $10100$ , $1000100$ , $10000010$ , $100000001$ . $\square$

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Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.