# 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. ! Extra }, or forgotten $.)

## 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)