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1998 AHSME Problems/Problem 12

Problem

How many different prime numbers are factors of $N$ if

$\log_2 ( \log_3 ( \log_5 (\log_ 7 N))) = 11?$

$\mathrm{(A) \ }1 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ }3 \qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ }7$

Solution

Re-writing as exponents, we have $\log_3 ( \log_5 (\log_ 7 N)) = 2^{11}$, and so forth, such that $N = 7^{5^{3^{2^{11}}}}$, which only has $7$ as a prime factor $\mathbf{(A)}$.

See also

1998 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
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All AHSME Problems and Solutions

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