Question

If $\left\{x\right\}$ denotes the fraction part of ${}^{\prime }{x}^{\prime }$, then $\left\{\frac{3^{1001}}{82}\right\}=$

• A. $\frac{9}{82}$
• B. $\frac{81}{82}$
• C. $\frac{3}{82}$
• D. $\frac{1}{82}$

Answer 1

The correct option is A $\frac{9}{82}$

$\Rightarrow \left\{\frac{3^{1001}}{82}\right\}$

$=3^4\to 81$ $1-\frac{1}{82}-1$

$=\frac{(81{)}^{997}}{82}$ $\frac{81}{82}$

$=\frac{(82-1{)}^{997}}{82}$

$=\{-\frac{(1-82{)}^{997}}{82}\}$

${=}^{997}{C}_0(-82{)}^0{+}^{997}{C}_1\frac{(-82{)}^1}{82}$

$=-\frac{1}{82}$

All will in proper division of 82.

$=1-\frac{1}{82}-1$

$=\{\frac{81}{82}-1\}$

$=\frac{81}{82}$