Difference between revisions of "2008 AMC 12B Problems/Problem 23"
(New page: ==Problem 23== The sum of the base-<math>10</math> logarithms of the divisors of <math>10^n</math> is <math>792</math>. What is <math>n</math>? <math>\textbf{(A)}\ 11\qquad \textbf{(B)}\ ...) |
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Revision as of 22:50, 1 March 2008
Problem 23
The sum of the base- logarithms of the divisors of is . What is ?
Solution
Every factor of will be of the form . Logarithmically, addition and multiplication are interchangeable (i.e. ), so we need only count the number of 2's and 5's occurring in total. For every factor , there will be another , so it suffices to count the total number of 2's occurring in all factors (because of this symmetry, the number of 5's will be equal). And since , the final sum will be the total number of 2's occurring in all factors of .
There are choices for the exponent of 5 in each factor, and for each of those choices, there are factors (each corresponding to a different exponent of 2), yielding total 2's. The total number of 2's is therefore . Plugging in our answer choices into this formula yields 11 (answer choice A) as the correct answer.