Question 10

# If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is/are true?A. n is oddB. n is primeC. n is a perfect square

Solution

Let the four consecutive positive integers be $$a,a+1,a+2$$ and $$a+3$$.
Therefore, $$n=1+a(a+1)(a+2)(a+3)$$
Or, $$n = 1+(a^2+3a)*(a^2+3a+2)$$
Or, $$n = (a^2+3a)^2 + 2*(a^2+3a)+1 = (a^2+3a+1)^2$$
Hence, n is a perfect square and therefore not a prime.

The product of four consecutive positive integers is always even. Hence, n is always odd.
Therefore, from the given statements, only A and C are true.