1985 AIME Problems/Problem 10

Problem

How many of the first 1000 positive integers can be expressed in the form

$\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor$ ,

where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$ ?

Solution

We will be able to reach the same number of integers while $x$ ranges from 0 to 1 as we will when $x$ ranges from $n$ to $n + 1$ for any integer $n$ . Since $\lfloor 2\cdot50 \rfloor + \lfloor 4\cdot50 \rfloor + \lfloor 6\cdot50 \rfloor + \lfloor 8\cdot50 \rfloor = 100 + 200 + 300 + 400$ , the answer must be exactly 50 times the number of integers we will be able to reach as $x$ ranges from 0 to 1, including 1 but excluding 0.

As we change the value of $x$ , the value of our expression changes only when $x$ crosses rational number of the form $\frac{m}{n}$ , where $n$ is divisible by 2, 4, 6 or 8. Thus, we need only see what happens at the numbers of the form $\frac{m}{\textrm{gcd}(2, 4, 6, 8)} = \frac{m}{24}$ . This gives us 24 calculations to make; we summarize the results here: