2014 AMC 10A Problem 25

The number 5^{867} is between 2^{2013} and 2^{2014}. How many pairs of integers (m,n) are there such that 1\leq m\leq 2012 and 5^n<2^m<2^{m+2}<5^{n+1}?

\textbf{(A) }278\qquad\textbf{(B) }279\qquad\textbf{(C) }280\qquad\textbf{(D) }281\qquad\textbf{(E) }282\qquad

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.