Question

Find the highest power of $2$ contained in $150!$.

• A. $150$
• B. $146$
• C. $140$
• D. $156$

Answer 1

The correct option is B

$146$

Step 1: Stating concept used

If $p$is prime then the highest power of $p$ in a factorial $n$is given by

$\left[\frac{n}{p}\right]+\left[\frac{n}{p^2}\right]+\left[\frac{n}{p^3}\right]+\dots .......$ where $\left[.\right]$ represents just the integral part of the answer and ignores the fractional part.

Here, $150!=1\times 2\times 3\times 4\times 5\times 6\times 7\times 8\times ..............\times 148\times 149\times 150$

Step 2: Find the highest power of $2$ in $(150!)$:

The highest power of $2$ in $150!$ is

$\left[\frac{150}{2}\right]+\left[\frac{150}{2^2}\right]+\left[\frac{150}{2^3}\right]+\left[\frac{150}{2^4}\right]+\left[\frac{150}{2^5}\right]+\left[\frac{150}{2^6}\right]+\left[\frac{150}{2^7}\right]+\left[\frac{150}{2^8}\right]+.....$

$=\left[75\right]+[37.5]+[18.75]+[9.375]+[4.688]+[2.344]+[1.172]+[0.58]+....$

$$\begin{gathered} =75+37+18+9+4+2+1+0 \\ =\mathbf{146} \end{gathered}$$

Final answer:

Hence, the correct option is B.