What is the largest power of 2 that divides $200!/100!$.

Question

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 1

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.

Answer 2

HINT.-The largest power of the prime $p$ dividing $n!$ is given by the very known formula $$\lfloor\frac np\rfloor+\lfloor\frac {n}{p^2}\rfloor+\lfloor\frac {n}{p^3}\rfloor+\cdots$$