What multiples of $d$ are still multiples of $d$ when they have their digits reversed? I teach at a school for 11 to 18 year olds. Every term I put up a Challenge on the wall outside my classroom.
This question is one that I have devised for that audience. I think that it is quite interesting and I would like to share it with a wider audience. I am not aware of it being a copy of another question.
An underlying assumption to this question is that we are writing numbers in base 10. A further question could be to investigate the same question in other bases.
Define the reverse of a number as the number created by writing the digits of the original number in the opposite order. Leading zeroes are ignored, although you will see that this makes very little difference to my question if this stipulation is removed.
Thus $Reverse(1456)=6541$ and $Reverse(2100)=12$
My question is this: For a given value $d$, what numbers $x$ have the property that $x$ is a multiple of $d$ and $Reverse(x)$ is a multiple of $d$?
I call such numbers "reversible multiples of $d$."
Clearly there are "brute force" ways to investigate this question, but I am looking for more subtle answers.
A good answer should address the following:
a) For some values of $d$ all multiples are reversible multiples. List those values, with a proof of why this is so.
b) For other values there are certain properties that the multiple must have for it to be a reversible multiple. Explain these.
c) For at least one value there is an algorithm that can be used to construct reversible multiples. Describe such an algorithm.
Enjoy!

I think this question is fascinating, thanks for posting it.  In addition to $3$ and $9$, I might add that multiples of $11$ are also always reversible.  Therefore, multiplies of $33$ and $99$ are also always reversible.  One might hope that squaring an always reversible number leads to an always reversible number, but that's not true:  $81$ and $121$ multiples are not always reversible.  Similarly, $33^2 = 1089 \ $ is not always reversible. However, 1089 is one of my favorite numbers because it is reversible for the first ten multiples, which is pretty unusual for a 4 digit number!
Now, $1001$ is reversible for much more than just the first ten multiples.  It is reversible for so many multiplies that one might be tempted to conjecture it is always reversible.  However, a quick Mathematica search reveals $1001 \cdot 1009 \ $ is not reversible, even though it any multiple less than $1009$ is reversible!  
Is $1, 3, 9, 11, 33, 99 \ $ the complete list of numbers whose multiples are always reversible?  I do not know ...   (Edit:  actually, this question is answered here: https://oeis.org/A018282/a018282.pdf )
