Given n = 1 + x and x is the product of four consecutive integers. Then which of the following us true ?
I. n is an odd integer.
II. n is prime.
III. n is a perfect square

A. Only I is correct

B. Only III is correct

C. Both I and II are correct

D. Both I and III are correct

Answer: Option D

Solution(By Examveda Team)

Out of four consecutive integers two are even and therefore, their product is even and on adding 1 to it, we get an odd integer. So, n is odd. Some possible values of n are as under :
n = 1 + (1 × 2 × 3 × 4) = (1 + 24) = 25 = 5²
n = 1 + (2 × 3 × 4 × 5) = (1 + 120) = 121 = 11²
n = 1 + (3 × 4 × 5 × 6) = (1 + 360) = 361 = 19²
n = 1 + (4 × 5 × 6 × 7) = (1 + 840) = 841 = 29²
Ans so on.....
Hence, n is odd and a perfect square.