Powers of $2$ 's After Another

Number Theory Level 1

$n$	$\hspace{10mm} 2^n$	Concatenation of the powered numbers	Divisibility checked
0	$\hspace{5mm} 2^0=1$	$\hspace{25mm} 1$	$\hspace{8mm} 1\, \big\|\, 1$
1	$\hspace{5mm} 2^1=2$	$\hspace{25mm} 12$	$\hspace{8mm} 2\, \big\|\, 12$
2	$\hspace{5mm} 2^2=4$	$\hspace{25mm} 124$	$\hspace{8mm} 4\, \big\|\, 124$
3	$\hspace{5mm} 2^3=8$	$\hspace{25mm} 1248$	$\hspace{8mm} 8\, \big\|\, 1248$
4	$\hspace{5mm} 2^4=16$	$\hspace{25mm} 124816$	$\hspace{8mm} 16\, \big\|\, 124816$

As we get greater and greater numbers in column 3 of the table by concatenation (i.e. 12481632, 1248163264, ...) for $n>4,$ will the divisibility in the last column still hold?

Powers of 222's After Another

Powers of $2$ 's After Another