If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is true?
II. n is a prime number
III. n is a perfect square
IV. n is odd number
We know that product of any two consecutive positive integers is always even number.
So, X = product of any four consecutive positive integers is always even number.
Now, n = (1 + x), is alway a odd number
Let x = 220.127.116.11 = 24
n = 1 + 24 = 25, which is a perfect square
again, for, x = 18.104.22.168 = 120
n = 1 + 120 = 121, a perfect square.
And this will follow for all positive integer value.
Hence, n is a perfect square odd number.