Question

If $\frac{1}{2 \cdot 3^{10}}+\frac{1}{2^{2} \cdot 3^{9}}+\ldots \frac{1}{2^{10} \cdot 3}=\frac{ K }{2^{10} \cdot 3^{10}}$, then the remainder when $K$ is divided by $6$ is

• A. 1
• B. 2
• C. 3
• D. 5

Answer 1

Answer: (D)

$\frac{1}{2 \cdot 3^{10}}+\frac{1}{2^{2} \cdot 3^{9}}+\frac{1}{2^{3} \cdot 3^{8}}+\ldots .+\frac{1}{2^{10} \cdot 3}=\frac{K}{2^{10} \cdot 3^{10}}$
$K =2^{9}+2^{8} \cdot 3+2^{7} \cdot 3^{2}+\ldots \ldots+3^{9}$
$=\frac{2^{9}\left(\left(\frac{3}{2}\right)^{10}-1\right)}{\frac{3}{2}-1}=3^{10}-2^{10}$
Now, $3^{10}-2^{10}=\left(3^{5}-2^{5}\right)\left(3^{5}+2^{5}\right)$
$=(211)(275)$
$=(35 \times 6+1)(45 \times 6+5)$
$=6 \lambda+5$
Remainder is $5 .$