elementary number theory - Divisibility Tests for Palindromes?

Let $n$ be a positive integer. My question, loosely, is

• Which palindromes are divisible by $n$?

Of course, the question has the easy answer "those that are divisible by $n$" unless I specify in what terms I want to characterize the divisibility. I'm not quite sure myself of how to answer this, so I will give some examples and general thoughts here, which will hopefully convey the spirit of the question.

Example of what I'm looking for:

• Every palindrome with an even amount of digits is divisible by $11$

Example of what I'm not looking for:

• A non-zero palindrome is divisible by $5$ if and only if it starts with a $5$

Both of these examples are found by applying some well known divisibility rules. The example of what I'm looking for is the case of the divisibility rule for $11$, which lets on simply check the alternating sum of the digits. But the nice thing is that one doesn't need to know any specific digits of the palindrome; only its length and the fact that it is a palindrome are needed. It is also a little less obvious than the example of what of I'm not looking for (at least to me.) Are there any other of these "cute" rules which let one easily see that a palindrome is or is not divisible by a certain integer? Has anyone ever studied this question in more detail?

I understand that the question of checking whether a certain number is divisible by another is already hard by itself, so maybe the question should be rephrased as

• For which integers $n$ and $P$ we can use the fact that $P$ is palindromic to our advantage in determining whether $n$ divides $P$?

Finally, as a sort of bonus question if anything in this direction is possible, I am particularly interested in the case in which $n$ is itself palindromic, i.e. determining whether a palindrome possesses a palindromic factor.

Thanks in advance for any thoughts whatsoever on the issue.

Answer

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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