2014 AMC 12B Problems/Problem 15
Revision as of 19:14, 20 February 2014 by Kevin38017 (talk | contribs) (Created page with "==Problem== When <math>p = \sum\limits_{k=1}^{6} k \ln{k}</math>, the number <math>e^p</math> is an integer. What is the largest power of 2 that is a factor of <math>e^p</math>...")
Problem
When , the number
is an integer. What is the largest power of 2 that is a factor of
?
$\textbf{(A)}\ 2^{12}\qquad\textbf{(B)}\ 2^{14}\qquad\textbf{(C)}\ 2^{16}\qquad\textbf{(D)}}\ 2^{18}\qquad\textbf{(E)}\ 2^{20}$ (Error compiling LaTeX. Unknown error_msg)
Solution
Let's write out the sum. Our sum is equal to
Raising
to the power of this quantity eliminates the natural logarithm, which leaves us with
This product has
powers of
in the
factor,
powers of
in the
factor, and
powers of
in the
factor.
This adds up to
powers of two which divide into our quantity, so our answer is
(Solution by kevin38017)