Question

The largest power of $2$ that divides $\frac{200!}{100!}$ is

• A. $98$
• B. $99$
• C. $100$
• D. $101$

Answer 1

The correct option is C $100$

Largest power of $2$ in $200!$ by Legendre's formula

$=[\frac{200}{2}]+[\frac{200}{4}]+[\frac{200}{8}]\cdots +[\frac{200}{128}]$ $=100+50+25+12+6+3+1=197$

Similarly, largest power of $2$ in $100!$ is

$=[\frac{100}{2}]+[\frac{100}{4}]+\cdots +[\frac{100}{64}]=50+52+12+6+3+1=97$

Thus, Largest power of $2$ in $\frac{200!}{100!}$ is $=197-97=100$