MCQ
The largest power of $2$ that divides $\frac{200 !}{100 !}$ is
- A$98$
- B$99$
- ✓$100$
- D$101$
Exponent of $2$ in $200 !$.
$=\left[\begin{array}{c}200 \\ 2\end{array}\right]+\left[\begin{array}{c}200 \\ 2^2\end{array}\right]+\left[\begin{array}{c}200 \\ 2^3\end{array}\right]+\left[\begin{array}{c}200 \\ 2^4\end{array}\right]+\left[\begin{array}{c}200 \\ 2^5\end{array}\right] +\left[\frac{200}{2^8}\right]+\left[\begin{array}{c}200 \\ 2^7\end{array}\right]+\left[\begin{array}{c}200 \\ 2^8\end{array}\right]$
$=100+50+25+12+6+3+1=197$
Exponent of $2$ in $100!$
$=\left[\frac{100}{2}\right]+\left[\frac{100}{2^2}\right]+\left[\begin{array}{c}100 \\ 2^3\end{array}\right]+\left[\begin{array}{c}100 \\ 2^4\end{array}\right]+\left[\begin{array}{c}100 \\ 2^5\end{array}\right]+\left[\begin{array}{c}100 \\ 2^6\end{array}\right]+\left[\begin{array}{c}100 \\ 2^7\end{array}\right]$
$\operatorname{In} \frac{200 !}{100 !}=\frac{2^{197}}{2^{97}}=2^{100}$
$\therefore$ The largest power of $2$ is $100$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$1.$ Which of the following is correct?
$(A)$ $a_{17}=a_{16}+a_{15}$ $(B)$ $c_{17} \neq c_{16}+c_{15}$
$(C)$ $b_{17} \neq b_{16}+c_{16}$ $(D)$ $a_{17}=c_{17}+b_{16}$
$2.$ The value of $b_6$ is
$(A)$ $7$ $(B)$ $8$ $(C)$ $9$ $(D)$ $11$
Give the answer question $1$ and $2.$