Problem

Let n be a natural number. Prove that

\displaystyle\bigg\lfloor\frac{n}{1}\bigg\rfloor +\bigg\lfloor\frac{n}{2}\bigg\rfloor +\bigg\lfloor\frac{n}{3}\bigg\rfloor +\ldots +\bigg\lfloor\frac{n}{n}\bigg\rfloor + \lfloor\sqrt{n}\rfloor

is even. (\lfloor x\rfloor is the floor, or the whole part, function.)

Solution

I placed two solutions on a separate page.