Consider the following two statements : I. if n is a composite number, then n divides n - 1 ! II. There are infinitely many natural numbers n such that n 3 + 2 n 2 + n divides n ! . Then

Step-by-step solution

If n is a composite number (Take n = 4 ) For n = 4 , n does not divide n - 1 ! Hence Ist statement is false (II) n 3 + 2 n 2 + n = n n + 1 2 n n + 1 2 divides n ! If n is such that n + 1 is a prime number (Take n = 6 ) so n n + 1 2 does not divide n ! but there are infinite values of n = 104 , 109 , 114 , … for which n n + 1 2 divide n ! but it is not true for every natural numbers.