What is the largest power of 2 that divides $200!/100!$.
No use of calculator is allowed.
I had proceeded in a brute force method which i know regret..
I would like to know your methods.
Answer
Find highest power of $2$ in $200!$ and $100!$, using Legendre's formula
In $200!$, highest power of $2$
$$=\lfloor 200/2 \rfloor +\lfloor 200/4 \rfloor +\lfloor 200/8 \rfloor +\lfloor 200/16 \rfloor +\lfloor 200/32 \rfloor +\lfloor 200/64 \rfloor +\lfloor 200/128 \rfloor $$
$$=100+50+25+12+6+3+1=197$$
In $100!$, highest power of $2$
$$=\lfloor 100/2 \rfloor +\lfloor 100/4 \rfloor +\lfloor 100/8 \rfloor +\lfloor 100/16 \rfloor +\lfloor 100/32 \rfloor +\lfloor 100/64 \rfloor$$
$$= 50 + 25+12+6+3+1 =97$$
Now, just subtract the two, and we get $100$ as the answer.
No comments:
Post a Comment