Proof for divisibility rule for palindromic integers of even length I am studying for a test and came across this in my practice materials. I can prove it simply for some individual cases, but I don't know where to start to prove the full statement.

Prove that every palindromic integer in base $k$ with an even number of digits is divisible by $k+1$.


HINT  $\rm\ \ mod\ \ x+1:\ \ f(x) + x^{n+1}\:(x^n\ f(1/x))\ \equiv\ f(-1) - f(-1)\:\equiv\: 0$  
Remark $\ \ $ It is simple to verify that $\rm\ x^n\ f(1/x)\ $ is the reversal of a polynomial $\rm\:f\:$ of degree $\rm\:n\:,\:$ therefore the above is the general palindromic polynomial with even number of coefficients.  
See also the closely related question.
