There are inﬁnitely many prime numbers.
Proof. Following Euclid’s proof, we shall show that to every prime p there is a
greater one. Assume that p is the greatest prime number, and let q = 1 · 2 · 3 · . . . · p.
Then, q + 1 is not divisible by 2, 3, . . . , p. It follows that q is divisible only by 1
and itself, and thus, it is a prime number greater than p. This, however, contradicts
the hypothesis that p is the greatest prime, and it follows that there is no greatest
prime. In other words, the set of primes is inﬁnite.
Help clarify how 1*2*3..*p + 1 is not divisible by 1,2,3 etc..

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So q would be odd? How can we verify that in any instance it is not divisible by 3 for example?

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