When is $X/R(X)$ an integer where $R(X)$ is the reverse of an integer $X$? My question concerns reverse numbers (e.g. $1234 → 4321$).
Is it possible to find integer solutions greater than $1$ for such numbers when you take their ratio? I am not interested in trivial solutions such as powers of ten and their multiples ($0, 100, 20, 1100$ etc.).
Let's say you have a number $X$ and $R(X)$ so that $X/R(X) = n$.
I've done some testing and have not found any solutions for $n = 2,3$ up to $X = 10^7$.
I'm fairly certain that I have proved that such a number cannot exist if it has an odd number of digits.
Any ideas or solutions? (This is just an idea that occurred to me, nothing I really have to solve)
Thanks!
 A: Suppose we have $n*abcd=dcba$.
Just looking at the leftmost digit, we have $d=na+m$ where $m$ is the amount carried over from the digit to the right. Note that $m<n$.
Looking at the rightmost digit, we have $10p+a=nd$ where $p$ is the amount carried over to the left. Again we have $p<n$.
Combining these two equations you get $10p+a=n(na+m)$ which simplifies to $(n^2-1)a = 10p-nm$.
Case n=2:
The equation becomes $3a = 10p-2m$ with $p,m\in\{0,1\}$. This is easily verified to have no integer solutions.
Case n=3:
The equation becomes $8a = 10p-3m$, with $p,m\in\{0,1,2\}$. Again this is easily checked to have no solutions with $a$ being a single digit.
Note that this argument works for any length of numbers, so there are no solutions with $n=2$ or $n=3$.
A: 
Not a 'real' answer, but it was too big for a comment. I think that you're looking for a solution without using a calculator or PC but maybe this gives some insight. I did a quick search where I look for values of $n$ that satisfy $12\le n\le10^9$

I wrote and ran some Mathematica-code:
In[1]:=Clear["Global`*"];
ParallelTable[
  If[IntegerQ[n/IntegerReverse[n]]&&IntegerQ[n/10]==False&& 
    PalindromeQ[n]==False,n,Nothing],{n,12,10^9}]//.{}-> 
  Nothing

Running the code gives:
Out[1]={8712, 9801, 87912, 98901, 879912, 989901, 8799912, 9899901,
87128712, 87999912, 98019801, 98999901, 871208712, 879999912,
980109801, 989999901}

Where I removed the palindromic numbers and numbers of the form $10^k$ for $k\in\mathbb{N}$.
