Question

If {x} denotes the fractional part of x, then $\left\{\frac{3^{10001}}{82}\right\}$ is:

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

$\left\{\frac{3^{10001}}{82}\right\}\times 82$ is the remainder when we divide $3^{1001}$ by 82.(If we write $\left\{\frac{3^{10001}}{82}\right\}$ as 82 k+m,then m is the remainder we get when we divide $3^{1001}$ by 82. $\left\{\frac{3^{10001}}{82}\right\}\times 82$ is $\left\{\frac{82k+m}{82}\right\}$.The fractional part is $\frac{m}{82}$.So we just need to find the remainder we get when we divide $3^{10001}$ by 82)

We have to express $3^{10001}$ in terms of 82.We know $3^4$ = 81 = 82 - 1.We will start like this.

$3^{10001}=3(81{)}^{250}$

$3^{10001}=3(82-1{)}^{250}$

$3(82-1{)}^{250}=3[\left({}^{250}{C}_0\right(82{)}^{250}-{}^{250}{C}_1(82{)}^n+{}^{250}{C}_2(82{)}^{n-1}+............{}^{250}{C}_{248}(82{)}^2+{}^{250}{C}_{249}(82)+{}^{250}{C}_{250})]$

$3(82-1{)}^{250}$ = 3[(Multiple of 82+1)]

$3(82-1{)}^{250}$ = 3 $\times$ Multiple of 82+3

$\Rightarrow$ The remainder when we divide $3^{1001}$ by 82 is 3

$\Rightarrow$ $\left\{\frac{3^{10001}}{82}\right\}$ is $\frac{3}{82}$