Question

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 $5^n < 2^m < 2^{m+2} < 5^{n+1}?$

$278$

$279$

$280$

$281$

$282$

Solution:

Since $2^2<5<2^3$ , each interval $(5^n,5^{n+1})$ contains either two or three powers of $2$ . The desired inequality holds exactly for intervals containing three such powers.

For $0\le n<867$ , let $d$ be the number of intervals with two powers of $2$ , and let $t$ be the number with three powers of $2$ . Then $d+t=867$ .

Because $2^{2013}<5^{867}<2^{2014}$ , these intervals contain $2013$ powers of $2$ altogether, so $2d+3t=2013$ .

Solving the system gives $t=279$ .

Thus, B is the correct answer.

2014 AMC 10A Problem 25

Solution:

Problem 25 in Other Years