Question

If $\left\{x\right\}$ denotes the fractional part of $x$, then $\left\{\frac{2^{2003}}{17}\right\}$ is

• A. $\frac{2}{17}$
• B. $\frac{4}{17}$
• C. $\frac{8}{17}$
• D. $\frac{16}{17}$

Answer 1

The correct option is C $\frac{8}{17}$

$2^{2003}=2^3.{2^4}^{500}=8.{16}^{500}=8.{(17-1)}^{500}$

$\Rightarrow 2^{2003}=8[{\left(17\right)}^{500}-{{}^{500}C}_1{\left(17\right)}^{499}+....-{{}^{500}C}_{499}{\left(17\right)}^1+1]$

$\frac{2^{2003}}{17}=8k+\frac{8}{17}$ where $k={\left(17\right)}^{499}-{{}^{500}C}_1{\left(17\right)}^{498}+...+{{}^{500}C}_{499}$ such that $k$ is an integer.

Thus, $\left\{\frac{2^{2003}}{17}\right\}=\frac{8}{17}$