$\lfloor 2x \rfloor \lfloor 3x \rfloor$ [closed]

From $1$ to $10000$ including both, how many of those integers can be written as:

$$\lfloor 2x \rfloor \lfloor 3x \rfloor$$

Where $x$ is a real number?

closed as off-topic by Zev Chonoles, Daniel W. Farlow, TravisJ, graydad, Jonas MeyerMay 27 '15 at 3:31

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Let $f(x) = \lfloor 2x \rfloor \lfloor 3x \rfloor$

Important Fact: $\lfloor x \rfloor$ is a function that remains constant, and only changes its value (increases by 1) when $x$ crosses an integer, say $n$.

Therefore $\lfloor 2x\rfloor$ increases when $x$ crosses $\frac{n}{2}$ and $\lfloor 3x \rfloor$ increases when $x$ crosses $\frac{n}{3}$. $\lfloor 2x \rfloor \lfloor 3x \rfloor$ will increase when $x$ crosses $\frac{n}{6}$.

Another important fact: The floor function is neither odd nor even, so we need to take two cases separately : $x$ is non-negative, and $x$ is negative.

Case 1: $x$ is non-negative

Note that when $x$ is an integer $n$, $f$ will be equal to $6n^2$. So all numbers of the form $6n^2$ can be written in this form.

Now let's see how the value of $f$ changes as $x$ varies from $n$ to $n+1$. Since the period of $f$ is $\text{lcm}( \frac{1}{2} , \frac{1}{3}) = 1$, we can be sure that the same pattern will be followed as $x$ goes from $n+1$ to $n+2$.

As we saw above, $f$ only changes when $x$ crosses and integral multiple of $\frac{1}{6}$, we can divide the interval $[n, n+1)$ into $6$ equal intervals.

$$\begin{array}{|l|c|c|c|}\hline \text{Interval} & \lfloor 2x \rfloor & \lfloor 3x \rfloor & f(x) \\\hline \left[n, n + \frac{1}{6}\right) & 2n & 3n & 6n^{2} \\ \left[n + \frac{1}{6}, n + \frac{2}{6}\right) & 2n & 3n & 6n^{2}\\ \left[n + \frac{2}{6}, n + \frac{3}{6}\right) & 2n & 3n + 1& 6n^{2} + 2n\\ \left[n + \frac{3}{6}, n + \frac{4}{6}\right) & 2n + 1& 3n + 1& 6n^{2} + 5n + 1\\ \left[n + \frac{4}{6}, n + \frac{5}{6}\right) & 2n + 1& 3n + 2& 6n^{2} + 7n + 2\\ \left[n + \frac{5}{6}, n + 1\right) & 2n + 1& 3n + 2& 6n^{2} + 7n + 2\\\hline \end{array}$$

We see that there are $4$ distinct values that $f$ takes as $x$ goes from $n$ to $n + 1$ viz. $6n^2, 6n^2 +2n, 6n^2 + 5n + 1, 6n^2 + 7n + 2$.

When $n = 0$, we only get two values - $1, 2$ since we don't want $0$. For all $n > 0$, we get $4$ values, for example for $n = 1$, we get $6, 8, 12$ and $15$. These are the numbers that can be written in the desired form. If we arrange them in ascending order, it is not hard to prove that $f(n) = 6n^2$ will be the $(4n - 1)^{\text{th}}$ term in the sequence.

$1$ is the minimum number in the range of $f$. To find the maximum number less than $10000$, the easiest way would be to find the greatest possible $n$, by setting $6n^2 \leq 10000$.

$$\implies n^2 \leq 1666$$ $$\implies n \leq 40$$ The maximum $n$ is $40$.

Now we check if $f(40 + \frac{5}{6}) < 10000$.

$f(40 + \frac{5}{6}) = 6 \times 40^2 + 7 \times 40 + 2 = 9882 \leq 10000$ which works as well.

We know that $f(40) = 6 \times 40^2 = 9600$ is the $(4 \times 40 - 1)$th term, ie $159$th term. Therefore $9882$ will be the $159 + 3 = 162$th term.

However, we mustn't forget Case 2, in which $x$ is negative.

Case 2: $x$ is negative.

For some negative integer $k$, let $m = k - \frac{1}{2}$. The motivation to do this will become clear shortly. Now let's see how $f$ varies as $x$ goes from $m$ to $m + 1$.

$$\begin{array}{|l|c|c|c|}\hline \text{Interval} & \lfloor 2x \rfloor & \lfloor 3x \rfloor & f(x) \\\hline \left[m + \frac{5}{6}, m\right) & 2k & 3k + 1 & 6k^{2} + 2k\\ \left[m + \frac{4}{6}, m + \frac{5}{6}\right) & 2k & 3k & 6k^{2}\\ \left[m + \frac{3}{6}, m + \frac{4}{6}\right) & 2k & 3k & 6k^{2}\\ \left[m + \frac{2}{6}, m + \frac{3}{6}\right) & 2k - 1& 3k - 1& 6k^{2} - 5k + 1\\ \left[m + \frac{1}{6}, m + \frac{2}{6}\right) & 2k - 1& 3k - 1& 6k^{2} - 7k + 2\\ \left[m, m + \frac{1}{6}\right) & 2k - 1& 3k - 2& 6k^{2} - 7k + 2\\\hline \end{array}$$

Since $k$ is a negative number, we can replace it by $- n$, where $n$ is a positive number. Observe for each $n$, the outputs are now $6n^2 - 2n, 6n^2 , 6n^2 + 5n + 1, 6n^2 + 7n + 2$.

That is, for each $n$, $f(- n)$ has exactly one element in its range $(6n^2 - 2n)$ which is not present in range of $f(n)$, ie a one-to-one correspondence.

Since we calculated $n$ to be $40$ in the previous case, there will be $40$ more numbers which can be written in this form if $x$ is negative.

There are $162 + 40 = 202$ integers from $1$ to $10000$ which can be written as $\lfloor 2x \rfloor \lfloor 3x \rfloor$.

If $n \le x < n+\dfrac{1}{3}$ for some integer $n$, then $\lfloor 2x \rfloor \lfloor 3x \rfloor = 2n \cdot 3n = 6n^2$.
If $n+\dfrac{1}{3} \le x < n+\dfrac{1}{2}$ for some integer $n$, then $\lfloor 2x \rfloor \lfloor 3x \rfloor = 2n(3n+1) = 6n^2+2n$.
If $n+\dfrac{1}{2} \le x < n+\dfrac{2}{3}$ for some integer $n$, then $\lfloor 2x \rfloor \lfloor 3x \rfloor = (2n+1)(3n+1) = 6n^2+5n+1$.
If $n+\dfrac{2}{3} \le x < n+1$ for some integer $n$, then $\lfloor 2x \rfloor \lfloor 3x \rfloor = (2n+1)(3n+2) = 6n^2+7n+2$.
Now, you need to figure out how many integers between $1$ and $10000$ can be written in one of these forms.