Question

Fractional part $\frac{2^{78}}{31}$ is

• A. $\frac{2}{31}$
• B. $\frac{8}{31}$
• C. $\frac{4}{31}$
• D. $\frac{10}{31}$

Answer 1

Answer: (B)

$\frac{8}{31}$

Sum of Binomial coefficients-

$(x+a{)}^n={}^n{C}_0{x}^n{a}^0+{}^n{C}_r{x}^{n-1}a+{}^n{C}_2{x}^{n-2}{a}^2+...$

$x=a=1:$

$\therefore$ $C_0+C_1+C_2+C_3+....=2^n$

$2^{78}$ can be written as :

$2^{78}=2^3{.2}^{75}=2^3(2^5{)}^{15}$

$2^3.(31+1{)}^{15}=8(31+1{)}^{15}$

Now,

$\Rightarrow$ $\frac{8}{31}(31+1{)}^{15}=\frac{8}{31}(31\lambda +1)$

$=8\lambda +\frac{8}{31}$

Since, $8\lambda$ is a integer and $\frac{8}{31}$ is a fractional part.