If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is/are true ?
I. n is odd
II. n is prime
III. n is a perfect square

Solution (By Examveda Team)

Let n = 1 + x = 1 + m (m + 1) (m + 2) (m + 3), where m is a positive integer.
Then, clearly two of m, (m + 1), (m + 2), (m + 3) are even and so their product is even.
Thus, x is even and hence n = 1 + x is odd
Also, n = 1 + m (m + 3) (m + 1) (m + 2) = 1 + (m² + 3m) (m² + 3m + 2)
⇒ n = 1 + y (y + 2), where m² + 3m = y
⇒ n = 1 + y² +2y = (1 + y)² , which is a perfect square.
Hence, I and III are true.