What is the greatest 5-digit palindrome for 7n to be a 6-digit palindrome?

[anemone]

Here is this week's POTW:

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What is the greatest 5-digit palindrome $n$ such that $7n$ is a 6-digit palindrome?

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[anemone]

Congratulations to Opalg for his correct and insightful solution, which you can find below:

Spoiler

We are looking for a number $n$ of the (decimal) form $abcba$ such that $7n$ is of the form $xyzzyx$.

Suppose first that $a = 9$. Since $7\times9 = 63$ it follows that the last digit of $7n$ is $3$ and the first digit of $n$ is at least $6$. So $7n$ cannot be a palindrome in this case.

Suppose next that $a=8$ and that $b=9$. Then $n = 89098 + 100c$. Therefore $7n = 623686 + 700c$. The addition of $700c$ can only affect the middle two digits of $7n$, so it cannot increase the $2$ (the second digit of $7n$) to an $8$. So $7n$ cannot be a palindrome.

Now suppose that $a=b=8$, so that $n$ is of the form $88088 + 100c$. Then $7n = 616616 + 700c$. But $616616$ is palindromic, and if $c\ne0$ then the addition of $700c$ will make the middle two digits unequal and destroy the palindrome.

So $n=88088$ is the largest 5-digit palindrome such that $7n$ is also a palindrome.