Difference between revisions of "2014 AMC 12B Problems/Problem 15"

Problem

When $p = \sum\limits_{k=1}^{6} k \ln{k}$, the number $e^p$ is an integer. What is the largest power of 2 that is a factor of $e^p$?

$\textbf{(A)}\ 2^{12}\qquad\textbf{(B)}\ 2^{14}\qquad\textbf{(C)}\ 2^{16}\qquad\textbf{(D)}}\ 2^{18}\qquad\textbf{(E)}\ 2^{20}$ (Error compiling LaTeX. ! Extra }, or forgotten \$.)

Solution

Let's write out the sum. Our sum is equal to $$1 \ln{1} + 2 \ln{2} + 3 \ln{3} + 4 \ln{4} + 5 \ln{5} + 6 \ln{6} =$$ $$\ln{1^1} + \ln{2^2} + \ln{3^3} + \ln {4^4} + \ln{5^5} + \ln {6^6} =$$ $$\ln{(1^1\times2^2\times3^3\times4^4\times5^5\times6^6)}$$ Raising $e$ to the power of this quantity eliminates the natural logarithm, which leaves us with $$e^p = 1^1\times2^2\times3^3\times4^4\times5^5\times6^6$$ This product has $2$ powers of $2$ in the $2^2$ factor, $4*2=8$ powers of $2$ in the $4^4$ factor, and $6$ powers of $2$ in the $6^6$ factor. This adds up to $2+8+6=16$ powers of two which divide into our quantity, so our answer is $\boxed{\textbf{(C)}\ 2^{16}}$