Question

If $\left\{x\right\}$ represents the fractional part of $x,$ then $\left\{\frac{5^{200}}{8}\right\}$ is

• A. $\frac{1}{4}$
• B. $\frac{1}{8}$
• C. $\frac{3}{8}$
• D. $\frac{5}{8}$

Answer 1

The correct option is B $\frac{1}{8}$

$\frac{5^{200}}{8}=\frac{{\left(5^2\right)}^{200}}{8}=\frac{(25{)}^{100}}{8}=\frac{(1+24{)}^{100}}{8}=\frac{1+{}^{100}{C}_1\cdot 24+\dots +{}^{100}{C}_2(24{)}^{100}}{8}=\frac{1+24k}{8},k\in N=\frac{1}{8}+3k\therefore \left\{\frac{5^{200}}{8}\right\}=\frac{1}{8}$