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 $\mathbf{(A)}$ .

1998 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
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1998 AHSME Problems/Problem 12

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