Question

The value of $\left\{\frac{3^{2003}}{28}\right\},$ where $\{.\}$ denotes the fractional part function, is

• A. $\frac{19}{36}$
• B. $\frac{19}{28}$
• C. $\frac{17}{36}$
• D. $\frac{17}{28}$

Answer 1

The correct option is B $\frac{19}{28}$

$3^{2003}=3^{2001}\cdot 3^2$
$=9(27{)}^{667}$
$=9(28-1{)}^{667}$
$=9[{{}^{667}{C}_{0}28}^{667}-{{}^{667}{C}_{1}28}^{666}+\cdots {+}^{667}{C}_{666}.(28{)}^1-{}^{667}{C}_{667}]$

i.e., if we divide $3^{2003}$ by $28,$ the remainder is $19$
$\therefore \left\{\frac{3^{2003}}{28}\right\}=\left\{\frac{19}{28}\right\}$