(In reply to How it could be done with a calculator (spoiler)
I have got the same 2 last digits by EXCEL column , starting with
a column of 23 numbers 2,3,...24 with 16 replaced by 1 and 5,10,15,20 by 1,2,3,4.
Going down with p(n)=MOD(n*p(n-1),100) resulted with 36.
Answer: the last two digits are 6 and 4.