Let PP Pn be primes Show that P P Pn 1 is divisible by

Let P,P, . . . Pn be primes. Show that P P ... Pn + 1 is divisible by none of the primes.

Solution

All numbers can be written as a product of primes, Because of the unique factorization theorem. This means your \"non-prime number\" can be written as a product of primes.

But it can\'t be written as a product of primes by definition, so it must be divisible by only itself - making your \"non-prime number\" a prime number.

Hence if P,P, . . . P_n be primes, then P P ... P_{n + 1} is divisible by none of the primes.