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Primes and their distribution

Definition: An integer p>1 is called prime number, simple prime, if its only positive divisors are 1 and p. An integer greater than 1 that is not prime is termed composite.

e.g: 2 ,3, 5, 7 are all primes while 4, 6, 8, 10 are composite number.

Theorem: If p is a prime and p\ab then p\a or p\b.

Proof: If p\a, then we need go no further, so let us assume that p∤ a, since the only positive divisors of p are 1 and p itself, this implies that gcd(p, a) = p . (in general, gcd(p, a) = p or gcd(p, a) = 1according as p\a or p∤ a.) Hence, citing Euclid’s lemma, we get p\b.     (Proved)

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