$X$ is set of number of form $a_{2k}10^{2k}+...+a_{o}$ as problem. lemma:For all k integer we have 2^k is divise a number belong to X. By induction. Return problem. Let $n=2^{p}.k$ (k is not divise by 5)and k odd.we will find a number in X which divise n. I see there is m in X so $2^{p}|m$ and supose m has d digits.Choose $f=10^{d+1}-1$ Since $10^{\phi(fk)}=1$$mod$ $fk$ then $\frac{m(10^{(d-1)\phi(fk)})}{10^{d+1}-1}$ is number which we need find.

cuopbientcd wrote: $X$ is set of number of form $a_{2k}10^{2k}+...+a_{o}$ as problem. lemma:For all k integer we have 2^k is divise a number belong to X. By induction. Return problem. Let $n=2^{p}.k$ (k is not divise by 5)and k odd.we will find a number in X which divise n. I see there is m in X so $2^{p}|m$ and supose m has d digits.Choose $f=10^{d+1}-1$ Since $10^{\phi(fk)}=1$$mod$ $fk$ then $\frac{m(10^{(d-1)\phi(fk)})}{10^{d+1}-1}$ is number which we need find. Why $\frac{m(10^{(d-1)\phi(fk)})}{10^{d+1}-1}$ is number which we need ? Can you explain more ?