Question

2. If p be a prime, p > 3 and let x be the product of
natural numbers 1, 2, 3, ..., (p-1), then consider the
following statements :
1. x is a composite number divisible by p.
2. x is a composite number not divisible by p, but
some prime greater than p may divide x.
3. x is not divisible by any prime (p - 2).
4. All primes less than (p-1) divide x.​

Answer 1

Given : p be a prime, p > 3 and let x be the product of   natural numbers 1, 2, 3, ..., (p-1),

To Find : Which statement is true:

1. x is a composite number divisible by p.

2. x is a composite number not divisible by p, but

some prime greater than p may divide x.

3. x is not divisible by any prime (p - 2).

4. All primes less than (p-1) divide x.​

Solution:

p  is a prime greater than 3

hence p is an odd number

only even prime number is 2  but not greater than 3

x = product of  1 , 2 , 3................, (p-1)

as p is a prime number and x has all factor less than p

hence x is not divisible by p   and also not divisible by any prime greater than p

Hence  x is a composite number divisible by p. - FALSE

x is a composite number not divisible by p, but  some prime greater than p may divide x. - FALSE

x = (p-1) ! =  (p-1)(p-2)(p-3)!  

Hence  x is divisible by any prime (p - 2).

x is not divisible by any prime (p - 2) - FALSE

All primes less than (p-1) divide x.​

TRUE      as x has all factors from  1 to (p - 1)

All primes less than (p-1) divide x.​ is only true statement

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