# Number system for operation

The last digit of some number 𝑋 in base 𝑘 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?

• I don't understand the first sentence. Did you mean to say, "The last digit of some number $X$ in base $k$ is $2?$" – saulspatz Feb 9 at 17:22
• Correct, I'll edit that... – Hmmman Feb 9 at 17:23
• I think you misunderstand the question. What they are asking is, for what values of $k$ is it true that if the last digit of a number $X$ in base $k$ is $2,$ then the last last digit of of $12_{10}X$ in base $k$ is $4.$ For example, this is true if $k=10.$ – saulspatz Feb 9 at 17:41
• It seems like you are right... but how can I determine all such systems? – Hmmman Feb 9 at 17:56
• And to be sure that it's correct for all digits, for example k = 10: 2, 12*2 = 24 how can i be sure that for ...2, 12*...2 = ....4 – Hmmman Feb 9 at 17:57

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 $$5,10,20$$.