I am trying to understand a proof that shows that

for any positive integer k and where is Euler's totient function.

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

there are at most 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 primes.

Any help would be greatly appreciated.

Thank you.