Question

Let $E=[\frac{1}{3}+\frac{1}{50}]+[\frac{1}{3}+\frac{2}{50}]+.......+$ upto $50$ terms, where is greatest integer function. The exponent of $2$ in $\left(E\right)!$ is

• A. $13$
• B. $15$
• C. $17$
• D. $19$

Answer 1

The correct option is B $15$

$E=[\frac{1}{3}+\frac{1}{50}]+[\frac{1}{3}+\frac{2}{50}]+...\dots +[\frac{1}{3}+\frac{50}{50}]$

$\frac{1}{3}+\frac{2}{3}=1$

$\frac{33}{50}<\frac{2}{3}<\frac{34}{50}$

$[\frac{1}{3}+\frac{33}{50}]=0;[\frac{1}{3}+\frac{34}{50}]=1$

From $34$ to $50$, there are $50-34+1=17$ terms.

$\therefore E=17$.

For ${171}^o$, the highest power of $2$ is given by

$\left[\frac{17}{2}\right]=8$

$\left[\frac{8}{2}\right]=4$

$\left[\frac{4}{2}\right]=2$

$\left[\frac{2}{2}\right]=1$

$8+4+2+1=15$

$2^{15}/17!$.