I am trying to understand a proof that shows that

$\displaystyle \frac{\pi(x)}{x}\leq\frac{\phi(k)}{k}+\frac{2k}{x}$

for any positive integer k and where $\displaystyle \phi(k)$ is Euler's totient function.

The proof comes to the point where it says among the integers

$\displaystyle k+1,k+2,\dots,2k$

there are at most $\displaystyle \phi(k)$ primes. The reasoning is that "since any integer not relatively prime to k has a prime factor in common with k that is less than or equal to k."

I understand that the reasoning is true, but I don't see how it explains that there are at most $\displaystyle \phi(k)$ primes.

Any help would be greatly appreciated.

Thank you.