How to understand Euclid's theorem on an infinite set of prime numbers?

Question

Mathematics

I can't understand this theorem. Here is the proof text:

Proof by contradiction. Let's assume that there is a finite set of primes, that is, there is the largest prime, let's call it P. We multiply all prime numbers from 2 to P and add 1:
2*3*5*7*11*...*P+1=M .
This number is not divisible by 2, by 3, by 5, by 7, ... by P, since there is always a remainder of 1. This means that either the number M is prime (but it is greater than the "largest" prime number P - a contradiction!), or it is composite, then it must be divisible by some prime number greater than P, but this also contradicts the assumption.

But how about this fact:

This number is not divisible by 2, by 3, by 5, by 7, ... by P, since there is always a remainder of 1.

made these two conclusions:

This means that either the number M is prime (but it is greater than the "largest" prime number P - a contradiction!)

or it is composite, then it must be divisible by some prime number greater than P, but this also contradicts the assumption.

??? I don't see a causal relationship.

Answer 1

Any natural number greater than 1 has a prime divisor.
M > 1, therefore, has a prime divisor, which cannot coincide with any of the components of the considered product, since it is not divisible by any of them.

Answer 2

It's clearer on the wiki:
And further on the link:

The fundamental theorem of arithmetic states that any composite number can be factored into a product of prime factors

Answer 3

It is assumed that there are a limited number of prime numbers in total, and the value of each of them is known to us: 2, 3, 5, 7, and so on up to the largest prime number P - and that’s it, there are no more prime numbers. All other (even very large) natural numbers greater than P are composite numbers - they can be represented as a product of a certain number of these prime numbers, each of which can be taken a certain number of times ...
Consider the number M: it is greater than P, and then, based on what was said above, it must be composite. But then M must be divisible by at least one prime number from our set of known primes without a remainder, and this is not the case. Therefore, the original statement is false. And it can be wrong in two ways: or M is still composite, but between P and M there is one or more prime numbers greater than P, by which M is divisible without a remainder (for example, 2*3*5*7*11* 13*17 + 1 = 510511 = 19*97*227 are examples of such numbers), or the number M itself is prime (for example, 2*3*5*7 + 1 = 211 is the 47th prime number), which, by the way, is completely does not mean that there are no other primes between P and M.
In any case, we find a prime number greater than the "greatest prime" known to us P, and this number itself becomes the "greatest prime number" - and since a similar operation can be done with any "greatest prime number", it turns out that "the largest prime number number" does not exist, in other words, the number of prime numbers is infinite.