Question

$\left\{\frac{2^{403}}{15}\right\}=\frac{k}{15}$ then find $k.$ (where $\text{'}\{.\}\text{'}$denotes fractional part function)

• A. $2$
• B. $8$
• C. $1$
• D. $4$

Answer 1

The correct option is B

$8$

Explanation for the correct option:

Find the values of $k$

We have been given an expression $\left\{\frac{2^{403}}{15}\right\}=\frac{k}{15}$.

We need to find the value of $k.$

We know,

$$\begin{gathered} \Rightarrow 2^4=1(mod15) \\ \Rightarrow 2^{4\times 100}=1^{100}(mod15) \\ \Rightarrow 2^{400}=1(mod15) \\ \Rightarrow (2^{400}\times 2^3)=(1\times 2^3)(mod15) \\ \Rightarrow 2^{403}=8(mod15) \\ \Rightarrow \frac{{\mathbf{2}}^{\mathbf{403}}}{\mathbf{15}}\mathbf{=}\frac{\mathbf{8}}{\mathbf{15}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{} \\ \mathbf{\Rightarrow }\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{k}\mathbf{=}\mathbf{8} \end{gathered}$$

Therefore, option (B) is the correct answer.