The last digit of some number in base is 2. Last number of $12_{10}*X$ in the same system is 4. How many systems that are suitable for these conditions for any X.
I m really confused for example I take $3_{10}$ as X, so for any system last digit will be 3, so there are no such systems?
Let me get you started. First of all, we must have $k>4$ or we couldn't have the digit $4$. Then the last digit of $X$ in base $k$ is $2$ if and only if $$X\equiv2\pmod{k}$$ Then we know that $$12X\equiv24\pmod{k}$$ and we need $$12X\equiv4\pmod{k}$$ in order for the last digit to be $4$.
So the question becomes, for what values of $k>4$ is it true that $$Y\equiv24\pmod{k}\implies Y\equiv4\pmod{k}$$
Well, if $k|(-4)$ and $k|(Y-24)$ then $k$ divides their difference, so $k|20$. The admissible $k$ are $5,10,20$.