# Difference between revisions of "2014 AMC 10A Problems/Problem 25"

The following problem is from both the 2014 AMC 12A #22 and 2014 AMC 10A #25, so both problems redirect to this page.

## Problem

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 $$5^n<2^m<2^{m+2}<5^{n+1}?$$ $\textbf{(A) }278\qquad \textbf{(B) }279\qquad \textbf{(C) }280\qquad \textbf{(D) }281\qquad \textbf{(E) }282\qquad$

## Solution

Between any two consecutive powers of 5 there are either 2 or 3 powers of 2 (because $2^2<5^1<2^3$). Consider the intervals $(5^0,5^1),(5^1,5^2),\dots (5^{866},5^{867})$. We want the number of intervals with 3 powers of 2.

From the given that $2^{2013}<5^{867}<2^{2014}$, we know that these 867 intervals together have 2013 powers of 2. Let $x$ of them have 2 powers of 2 and $y$ of them have 3 powers of 2. Thus we have the system

$x+y&=867$ (Error compiling LaTeX. ! Misplaced alignment tab character &.)
$2x+3y&=2013$ (Error compiling LaTeX. ! Misplaced alignment tab character &.)

from which we get $y=279$, so the answer is $\boxed{\textbf{(B)}}$.

(Solution by superpi83)