is there a natural number n for which (15)n is an even number 

Dear Student,
If you have asked for 15n, then for all even natural numbers 15n is an even number.
Example :
$$\begin{gathered} 15\times 2=30 \\ 15\times 4=60 \\ 15\times 6=90 \end{gathered}$$
all are even numbers
And if you have asked for ${15}^{\mathrm{n}}$ then it is always an odd number i.e. there is no natural number for which  ${15}^{\mathrm{n}}$ is an even number.

Regards