Given $2^n$, what is the largest power of $2$ that will divide any random concatenation of base $10$ digits of powers of $2$ ending with $2^n$?

Question

My first thought was that it would be $2^n$ itself, for example, if you concatenate $4$ and $2$ to get $42$, that's divisible by $2$ but not by $4$.

But whit $2^9 = 512$, you can concatenate $16$ and $512$ to get $16512$, which is divisible by $128$ but not by $256$ nor $512$. I half suspect (no pun intended) there is a concatenation ending in $512$ that is divisible by $4$ but not by $8$.

Does their exist a formula that will tell us what is the highest power of $2$ dividing any concatenation of powers of $2$ ending with $2^n$?

Answer 1

Hint: $2^n$ divides a number if and only if it divides the last $n$ digits of the number.