## Smith’s determinant

Let denote the greatest common divisor of and the Euler’s totient function. Prove that:

Solution

Discuss this at the forum of JoM here.

## Integral and Inequality

Let be a positive real valued and continuous function such that it is periodic of period . Prove that

Solution

Since the function is periodic , then it holds that:

Thus:

since forall .

## A sum on Beatty’s theorem

Let be positive irrational numbers such that . Evaluate the (pseudo) sum:

Solution

We are using Beatty’s theorem .In brief, it states that for positive irrational numbers with the sequences and are complementary. (i.e. disjoint and their union is ). Thus our sum is nothing else than

## No continuous mapping

Prove that there does not exist an and continuous mapping from to .

Solution

Due to connectness we have that where is an interval. Note that if we remove a point from the plane, it still remains connected. Having that in mind we observe that is connected and equals whereas this is not connected leading to a contradiction.