Question on Euclid's proof of the infinitude of primes

By the fundamental theorem of arithmetic, Q is prime or else it can be written as the product of

two or more primes. However, none of the primes pj divides Q, for if pj | Q, then pj divides

Q −p1p2 ···pn = 1. Hence, there is a prime not in the list p1, p2,...,pn. This prime is

either Q, if it is prime, or a prime factor of Q. This is a contradiction because we assumed that

we have listed all the primes. Consequently, there are infinitely many primes.

This is the proof provided in our textbook (Rosen) that demonstrates the infinitude of primes. I find myself unable to grasp this proof.

Specifically, when it is stated that "if pj | Q, then pj divides Q −p1p2 ···pn = 1." I don't understand why the first proposition implies the second. Can you help me with this?

**************************************************************************

....judging from what you didn't put in your quote from the book, you may have missed the part that sets up the proof by contradiction: Suppose that there are a finite number of primes; write them as p_1, p_2, ...  p_n.  Define the integer Q as  Q=p_1p_2...p_n+1.  Here's where what you quoted comes in:

By the fundamental theorem of arithmetic, Q is prime or else it can be written as the product of

two or more primes. However, none of the primes pj divides Q, for if pj | Q, then pj divides

Q −p1p2 ···pn = 1. Hence, there is a prime not in the list p1, p2,...,pn.

Really, you can see from the definition of Q that Q div p_j is the product of all of the p's except p_j, and the remainder Q mod p_j is 1.  This means that Q is not divisible by any of my finite list of primes, so it's not a composite number (i.e. a product of more than one prime), so it must be prime. But Q is also strictly bigger than any of my list of primes p_1, p_2, ..., p_n, so it is not one of p_1, p_2, ..., p_n.  But these were supposed to be all of the primes.   This is a contradiction.  Therefore my supposition that there are only a finite number of primes is false.  Therefore there are an infinite number of primes.