2018 AMC 12A Problem 7

Problem 7 of 25EasierAlgebra Number Theory

For how many (not necessarily positive) integer values of

n

is the value of

4000 \cdot \left(\tfrac{2}{5}\right)^n

an integer?

Show solution

Solution

Since

4000 = 2^5 \cdot 5^3,

the expression equals

2^{5+n} \cdot 5^{3-n}.

This is an integer exactly when both

5 + n \ge 0

and

3 - n \ge 0,

that is,

-5 \le n \le 3.

There are

3 - (-5) + 1 = 9

such integers. Thus, the correct answer is E.

Concepts: prime factorization · exponent

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.