What are all the positive integers $n$ such that $2^n$ ends in $n$? What are all the positive integers $n$ such that $2^n$ ends in $n$?


Example: $2^{36} = 68719476736$ ends in $36$.
Firstly, $n$ should be even.

Now


  $2^n=10i+4$ or $2^n=10i+6$


Unit digits agree when


  $n=20j-6$ or $n=20j+16$


that gives $2^{n}=100k+84$ or $100k+36$ respectively.

Since $20j-6 \pmod{ 4} \not \equiv  100k+84 \pmod{4}$,

we only need to try


  $n=100\ell+36$


Upcoming solutions are $736$, $8736$, $48736, \ldots$
Well last two digit possibilities 

04 08 16 32 64 28 56 12 24 48 96 92 84 68 36 72 44 88 76 52. 

Now we can check the results using a calculator.