Number of integers (less than $n$) that are divisible by a prime factor of $n$

Question

Given $n$ as $$n = \prod_{i = 1}^{m} {p_i}^{e_i} = {p_1}^{e_1} {p_2}^{e_2} \cdots {p_m}^{e_m}$$ where where the $p_i$ are distinct prime numbers.

My question is how the following statement is proved?

There are $\frac{n}{p_1}$ positive integers less than or equal to $n$ that are divisible by $p_1$.

Reference : https://artofproblemsolving.com/wiki/index.php?title=Euler%27s_totient_function

Answer 1

There are $\frac{n}{p_1}$ positive integers less than or equal to $n$ that are divisible by $p_1$, and those numbers are: $$ p_1, 2p_1,3p_1, \ldots,\frac n{p_1}p_1 $$ It's relatively easy to check that

• These are indeed multiplies of $p_1$
• They are all less than or equal to $n$
• They cover every possible (positive) multiple of $p_1$ up to and including $n$
• There are $\frac n{p_1}$ of them