question_answer

If the fractional part of the number \[\frac{{{2}^{\,403}}}{15}\] is \[\frac{k}{15}\], then k is equal to: [JEE Main Online Paper (Held On 09-Jan-2019 Morning]

A) 8     

B) 14    

C) 6     

D) 4

Correct Answer: A

Solution :

  \[\frac{{{2}^{403}}}{15}\,\,=\,\,{{2}^{3}}\,\,.\,\,\frac{{{2}^{400}}}{15}\,\,=\,\,\frac{{{2}^{3}}\,.\,({{2}^{4}})\,100}{15}\] \[=\,\,\frac{8{{[1+15]}^{100}}}{15}\] \[=\,\,\,\left\{ \frac{8.1}{15} \right\}\] \[\therefore \,\,k\,\,=\,\,8\]