I'm trying to prove the following property of Euler's totient function:
if $n \in \mathbb{N}$, then $$\varphi(n)=n\cdot \prod_{\substack{p|n\\p \ \text{prime}}} \left(1-\frac{1}{p}\right)$$ I have already proved that $\varphi(p^k)=p^k-p^{k-1}$ for $p$ prime and that the function is multiplicative. I've seen proofs where these two properties are used, but I'm having problems with the first step:
n has a unique prime factorization $n=p_1^{k_1}\cdots p_q^{k_q}$ with prime numbers $p_1<\ldots <p_q$ and $k_1,\ldots, k_q \in \mathbb{N}$. In order to use the multiplicative property on $n$, I have to prove that the numbers $p_1^{k_1}, \ldots , p_q^{k_q}$ are pairwise coprime. How do I do that? I.e. how would I prove that $p_1^{k_1}$ and $\left(p_2^{k_2}\cdots p_q^{k_q}\right)$ are coprime?