2014 AMC 10A Problem 25

What is being counted. The condition says $2^m$, $2^{m+1}$ and $2^{m+2}$ all fit strictly between $5^n$ and the next power of $5$. So we are counting gaps between neighbouring powers of $5$ that hold at least three powers of $2$.

Look at the first few gaps to get a feel for it:

1. Every gap holds two or three powers of $2$. Let $2^m$ be the first power of $2$ after $5^n$. The power of $2$ just before it is at most $5^n$, so $2^m$ is at most $2 \cdot 5^n$.
- Then $2^{m+1}$ is at most $4 \cdot 5^n$, which is less than $5 \cdot 5^n = 5^{n+1}$. So the gap holds at least two.
- And $2^{m+3} = 8 \cdot 2^m$, which is more than $8 \cdot 5^n$, which is more than $5^{n+1}$. So the gap holds at most three.
2. A gap with three powers gives exactly one pair $(m, n)$: $n$ is the gap, and $m$ is its first power of $2$. A gap with only two gives none.
3. Use the given fact. $5^{867}$ is between $2^{2013}$ and $2^{2014}$. So between $1 = 5^0$ and $5^{867}$ the powers of $2$ are exactly $2^1, 2^2, \ldots, 2^{2013}$. That is $2013$ of them.
4. Count the gaps. From $5^0$ to $5^{867}$ there are $867$ gaps: $5^0$ to $5^1$, $5^1$ to $5^2$, and so on up to $5^{866}$ to $5^{867}$. A power of $2$ is never a power of $5$ (other than $1$), so each of the $2013$ powers of $2$ sits in exactly one gap.
5. Set up an equation. Say $t$ gaps hold three powers and the other $867 - t$ hold two. Counting all the powers of $2$:
$$3t + 2(867 - t) = 2013.$$
The left side is $t + 1734$, so $t = 2013 - 1734 = 279$.
6. Check the limits on $m$. In these gaps the three powers are at most $2^{2013}$, so $m + 2 \le 2013$ and $m \le 2011$, which fits $m \le 2012$. Any gap beyond $5^{867}$ would need $2^m$ above $2^{2013}$, which breaks $m \le 2012$. So no other pairs exist.

The answer is $\boxed{\textbf{(B)}\ 279}$.

Since $5$ lies between $4$ and $8$, multiplying by $5$ jumps over either two or three doublings, never fewer and never more. So each gap is one of two types, and the total number of powers of $2$ decides how many of each. This is counting the same thing two ways: once directly ($2013$ powers of $2$) and once gap by gap ($3t + 2(867 - t)$). The odd-looking fact about $5^{867}$ is there to give both totals.