8.21 If $\log _{2}\left(\log _{3}\left(\log _{5}\left(\log _{7} N\right)\right)\right)=11$, then the number of distinct prime factors of $N$ is
(A) 1 .
(B) 2 .
(C) 3 .
(D) 4 .
(E) 7 .
(49th American High School Mathematics Examination, 1998)
[Solution] From the given, we have $\log _{3}\left(\log _{5}\left(\log _{7} N\right)\right)=2^{11}$, and $\log _{5}\left(\log _{7} N\right)=3^{2^{11}}$,
and $\log _{7} N=5^{3^{3^{11}}}$, thus $N=7^{5^{3^{2^{11}}}}$. Clearly, $N$ has only one prime factor, 7. Therefore, the answer is $(A)$.