Divisibility Tests for Palindromes?

An interesting case is $n=101$.

The $3$-digit palindrome $aba$ is divisible by 101 iff $b=0$.

The $4$-digit palindrome $abba$ is divisible by 101 iff $a=b$.

The $5$-digit palindrome $abcba$ is divisible by 101 iff $c=2a$.

The $6$-digit palindrome $abccba$ is divisible by 101 iff $a+b=c$.

The $7$-digit palindrome $abcdcba$ is divisible by 101 iff $d=2b$.

The $8$-digit palindrome $abcddcba$ is divisible by 101 iff $a+d=b+c$.

The $9$-digit palindrome $abcdedcba$ is divisible by 101 iff $e = 2(c-a)$.

The $10$-digit palindrome $abcdeedcba$ is divisible by 101 iff $a+b+e=c+d$.

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