Question 43
Let $n$ be a natural number. Prove that
$$
\left[\frac{n}{1}\right]+\left[\frac{n}{2}\right]+\left[\frac{n}{3}\right]+\cdots+\left[\frac{n}{n}\right]+[\sqrt{n}]
$$
is even. (Here $[x]$ denotes the largest integer smaller than or equal to $x$.)
$$
\left[\frac{n}{1}\right]+\left[\frac{n}{2}\right]+\left[\frac{n}{3}\right]+\cdots+\left[\frac{n}{n}\right]+[\sqrt{n}]
$$
is even. (Here $[x]$ denotes the largest integer smaller than or equal to $x$.)