I'm trying to understand Euclid's Theorem, using proof by contradiction, which states:
There are an infinite number of prime numbers.
In the book it has the following explanation: We assume that there are a finite number of prime numbers, $p_1, p_2, \dotsc, p_n$. We then consider an integer $Q$: $$Q:= p_1 \cdot p_2 \dotsb p_n+1$$
From the Fundamental Theorem of Arithmetic we know that any composite number can be represented as the product of various prime numbers. Therefore:
$$Q=p_1^{e_1} \cdot p_2^{e_2} \dotsb p_n^{e_n}, \ \ \text{for a suitable }e_1,\dotsc,e_n \in \mathbb{N_0}$$
Since $Q>1$, there is at least one $i \in [n]$ with $e_i \neq 0$. Therefore, for $p_i$ we have that:
$$p_i \mid Q \ \text{and} \ p_i \mid (Q-1)$$
This is a contradiction to our original assumption that $p_i \geq2$. Thus there are an infinite number of prime numbers.
I'm having difficulty understanding how the fact $p_i \mid Q \ \text{and} \ p_i \mid (Q-1)$ is used to come to the contradiction.
