Euler’s totient function

We define Euler’s totient function as follows:
$$\forall n \in \mathbb{N} ,\: \varphi (n)=\sum_{k=1}^{n}\mathbb{1}_{gcd(k,n)=1},$$meaning that $\varphi(n)$ equal the number of positive integer less (or equal) than n that are coprime with n.

1. Show that $\forall (a,b) \in \mathbb{N}^{2}, \: a|b \Rightarrow \varphi(a)| \varphi(b)$.

2. Show that $\forall (a,b) \in \mathbb{N}^2, \: gcd(a,b) =1\Rightarrow\varphi(a\cdot b)=\varphi(a)\cdot \varphi(b)$.

3. Then deduce that,

$$\varphi(n)=n\cdot \prod_{\substack{p|n\\p\in \mathbb{P}}}^n \left(1-\frac{1}{p}\right).$$

4. Moreover, show that $n =\sum_{d|n}\varphi(d)$.