How many positive integers $N \le 10^6$ such that $\lfloor \sqrt N \rfloor|N$?

Question

How many positive integers $N \le 10^6$ such that $N$ divided by $\lfloor \sqrt N \rfloor$?

I do not know how to solve this problem without a computer.

Answer 1

Let $a=\lfloor\sqrt N\rfloor$. We have $$a^2\le N<a^2+2a+1$$

Suppose that $a\mid N$. Then $$a\le\frac Na< a+2+\frac1a$$ So $N/a$ must be $a$, $a+1$ or $a+2$.

Therefore, the numbers that hold the statement are:

• The perfect squares
• The numbers of the form $a(a+1)$
• The numbers of the form $a(a+2)$

So you should solve these inequalitites: $$a^2\le 10^6$$ $$a(a+1)\le10^6$$ $$a(a+2)\le10^6$$

Note that these three sets of numbers (that is, the perfect squares, the numbers of the form $a(a+1)$ and the numbers of the form $a(a+2)$) are disjoint. (Why?)

Answer 2

Let $m=\lfloor\sqrt{N}\rfloor$. Then $m^2\leq N\leq m^2+2m$ and $N$ is a multiple of $m$.