Question

$\left\{\frac{2^{403}}{15}\right\}=\frac{k}{15}$. Then $k$ is

• A. $8$
• B. $9$
• C. $10$
• D. $7$

Answer 1

The correct option is A $8$

$\frac{2^{403}}{15}=\frac{{\left(2^4\right)}^{100}\times 2^3}{15}$

$\Rightarrow \frac{{(15+1)}^{100}\times 2^3}{15}$

$\Rightarrow {(15+1)}^{100}={15}^{100}+{15}^{99}\times \mathrm{1......}{1}^{100}$

$=15k+1$ ($k$ is an integer positive)

$\Rightarrow \frac{(15k+1)\times 8}{15}$

$\Rightarrow \frac{15\times 8\times k}{15}+\frac{8}{15}$

$\Rightarrow C+\frac{8}{15}$ ($C$ is a whole number)

$\left\{\frac{2^{403}}{15}\right\}=\{C+\frac{8}{15}\}=\frac{8}{15}$