2014 AMC 10A Problem 25

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

Each interval between consecutive powers of

5

contains two or three powers of

2

Use the total number of powers of

2

before

5^{867}

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.

Problem 25 in other years