Powers of 22's After Another

nn2n\hspace{10mm} 2^nConcatenation of the powered numbersDivisibility checked
020=1\hspace{5mm} 2^0=11\hspace{25mm} 111\hspace{8mm} 1\, \big|\, 1
121=2\hspace{5mm} 2^1=212\hspace{25mm} 12212\hspace{8mm} 2\, \big|\, 12
222=4\hspace{5mm} 2^2=4124\hspace{25mm} 1244124\hspace{8mm} 4\, \big|\, 124
323=8\hspace{5mm} 2^3=81248\hspace{25mm} 124881248\hspace{8mm} 8\, \big|\, 1248
424=16\hspace{5mm} 2^4=16124816\hspace{25mm} 12481616124816\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,n>4, will the divisibility in the last column still hold?

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