Question

If $\{p\}$ denotes the fractional part of the number $p,$ then $\left\{\frac{3^{200}}{8}\right\},$ is equal to

• A. $\frac{1}{8}$
• B. $\frac{5}{8}$
• C. $\frac{3}{8}$
• D. $\frac{7}{8}$

Answer 1

Answer: (A)

$\left\{\frac{3^{200}}{8}\right\}=\left\{\frac{{\left(3^2\right)}^{100}}{8}\right\}$
$=\left\{\frac{(1+8{)}^{100}}{8}\right\}$
$=\left\{\frac{1+{}^{100}{C}_1\cdot 8+{}^{100}{C}_2.8^2+\dots +{}^{100}{C}_{100}{8}^{100}}{8}\right\}$
$=\left\{\frac{1+8m}{8}\right\}$
$=\frac{1}{8}$