Prove that there are infinite number of primes.
Let there be finite number of primes a1, a2, a3,….an.
Here, a1< a2< a3….<an.
Now, let m = 1 + a1, a2, a3….an which is close to a1, a2, a3….an and is divisible by a1, a2, a3….an.
Here, m is a prime number or it has factors other than a1, a2, a3….an. Thus, there exists a positive prime number other than a1, a2,….an
Now, this point contradicts the fact that there are finite number of positive primes.
Therefore, there are infinite number of positive primes.