Question

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

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

Answer 1

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

$\left\{\frac{3^{200}}{8}\right\}=\left\{\frac{9^{100}}{8}\right\}=\left\{\frac{(8+1{)}^{100}}{8}\right\}$
$=\left\{\frac{{}^{100}{C}_0{+}^{100}{C}_1\left(8^1\right)1^{99}{+}^{100}{C}_2\left(8^2\right)1^{98}+...{+}^{100}{C}_{100}{8}^{100}}{8}\right\}$
$=\left\{\frac{{}^{100}{C}_0+8k}{8}\right\}$
$=\left\{\frac{1+8k}{8}\right\}=\{\frac{1}{8}+k\},k\in I$
$=\frac{1}{8}$