If the fractional part of the number 2^40315 is k15, then k is equal to:

Question

If the fractional part of the number $\dfrac{2^{403}}{15}$ is $\dfrac{k}{15}$, then $k$ is equal to:

Accepted Answer

$${{{2^{403}}} \over {15}}$$

$$ = {{{2^3}\, \cdot \,{2^{400}}} \over {15}}$$

$$ = {8 \over {15}}{\left( {16} \right)^{100}}$$

$$ = {8 \over {15}}{\left( {15 + 1} \right)^{100}}$$

$$ = {8 \over {15}}\left( {{}^{100}{C_0} + {}^{100}{C_1}\,15 + {}^{100}{C_2}{{\left( {15} \right)}^2} + .....{{\left( {15} \right)}^{100}}} \right)$$

$$ = {8 \over {15}} + 8\left( {{}^{100}{C_1} + {}^{100}{C_2}\,\left( {15} \right) + ..... + {{\left( {15} \right)}^{99}}} \right)$$

$$ = {8 \over {15}} + 8$$ (integer)

$$ \therefore $$  Fractional part $$ = {8 \over {15}}$$

According to the question,

$${k \over {15}} = {8 \over {15}}$$

$$ \Rightarrow $$   K $=$ 8

Solutions · 1