How many different prime numbers are factors of N if log_2(log_3(log_5(log_7N)))=11?

1998 AMC 12 Problem 12

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12.

How many different prime numbers are factors of

N

\log_2(\log_3(\log_5(\log_7N)))=11?

1

2

3

4

7

Undo the logarithms one at a time, starting with base

2

The final expression for

N

is a power of

7

Successively exponentiating gives

N=7^{\,5^{\,3^{\,2^{11}}}}.

Thus

7

is the only prime factor of

N,

so the correct answer is A.