. Show that there are infinitely many positive prime numbers.

1.	. Show that there are infinitely many positive prime numbers.
Answer» Let there be finite number of positive prime numbera1,a2,a3.......,an. Such thata1<a2<a3< .......an+<an. Letm= 1 + a1,a2,a3.......,an. It is near thata1,a2,a3.......,anis divisible bya1,a2,a3.......,an. mis a prime number or it has factors other thana1,a2,a3.......,an. There exits a positive prime number other thana1,a2,a3.......,an. This contradicts the fact that there are finite number of positive primes. Hence, there infinite number of positive primes.

Answer»

Let there be finite number of positive prime numbera1,a2,a3.......,an.

Such thata1<a2<a3< .......an+<an.

Letm= 1 + a1,a2,a3.......,an. It is near thata1,a2,a3.......,anis divisible bya1,a2,a3.......,an.

mis a prime number or it has factors other thana1,a2,a3.......,an. There exits a positive prime number other thana1,a2,a3.......,an.

This contradicts the fact that there are finite number of positive primes.

Hence, there infinite number of positive primes.

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