Smallest Multiple to "Palindromize"
Pyth, 10 9 bytes
&amp;eQf_I`*Q

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Explanation
&amp;eQf_I`*QT    Implicitly append T, and take input in Q. 
 eQ           Calculate input mod 10.
&amp;             Logical and, only evaluate the following if result ≠ 0.
   f          Starting from 1, find the first integer T for which... 
      `       ...the string representation of... 
       *QT    ...input multiplied by T... 
    _I        ...is equal to its reverse.