A) \[\frac{1}{15}\]
B) \[\frac{2}{15}\]
C) \[\frac{4}{15}\]
D) None of these
Correct Answer: A
Solution :
[a] \[\frac{{{2}^{4n}}}{15}=\frac{{{16}^{n}}}{15}=\frac{{{(1+15)}^{n}}}{15}\] \[=\frac{1+{{\,}^{n}}{{C}_{1}}\,15+{{\,}^{n}}{{C}_{2}}\,{{15}^{2}}+....+{{\,}^{n}}{{C}_{n}}\,\,{{15}^{n}}}{15}\] \[=\frac{1+15\,k}{15},\] where \[k\in N,=\frac{1}{15}+k\] \[\therefore \] Fractional part of \[\frac{{{2}^{4n}}}{15}\] is \[\frac{1}{15}\]
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