If p denotes the fractional part of the number p, then 3^2008 is equal to:

Question

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

Accepted Answer

$$\left\{ {{{{3^{200}}} \over 8}} \right\}$$

= $$\left\{ {{{{{\left( {{3^2}} \right)}^{100}}} \over 8}} \right\}$$

= $$\left\{ {{{{{\left( {1 + 8} \right)}^{100}}} \over 8}} \right\}$$

= $$\left\{ {{{1 + {}^{100}{C_1}.8 + {}^{100}{C_2}{{.8}^2} + .... + {}^{100}{C_{100}}{{.8}^{100}}} \over 8}} \right\}$$

= $$\left\{ {{{1 + 8K} \over 8}} \right\}$$

= $$\left\{ {{1 \over 8} + K} \right\}$$ where K $$ \in $$ Integer

$$ \therefore $$ Fractional part = $${{1 \over 8}}$$

Solutions · 1