Solve this following

Question:

If $\{p\}$ denotes the fractional part of the number

$\mathrm{p}$, then $\left\{\frac{3^{200}}{8}\right\}$, is equal to

 

  1. $\frac{1}{8}$

  2. $\frac{5}{8}$

     

  3. $\frac{3}{8}$

  4. $\frac{7}{8}$


Correct Option: 1

Solution:

$\left\{\frac{3^{200}}{8}\right\}=\left\{\frac{\left(3^{2}\right)^{100}}{8}\right\}$

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

$=\left\{\frac{1+{ }^{100} C_{1} \cdot 8+{ }^{100} C_{2} \cdot 8^{2}+\ldots+{ }^{100} C_{100} 8^{100}}{8}\right\}$

$=\left\{\frac{1+8 m}{8}\right\}$

$=\frac{1}{8}$

 

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