lastDigitsDec[n_]:=StringTake[StringTrim[StringSplit[ToString[BaseForm[n,10]]][[1]],("0")...],-10]

lastDigitsDec[4096!]

Table[4096*5^n,{n,0,12}]

Table[{n,20480*5^n,lastDigitsDec[(20480*5^n)!]},{n,0,1}]

lastDigitsDec[2560000!]

2560000/5^4