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 odd

B. n is prime

C. n is a perfect square

- A and C only.
- A and B only.
- A only.
- None of this.

x is the product of 4 consecutive positive integers, that means its even number (because there will be 2 even numbers and any positive integer multiplied by even gives even)

**This makes 'A' is True**. i.e. n is odd as even + 1 is odd

for 'B' lets put some values.

it is given that x is product of 4 consecutive positive integers.

Assume consecutive numbers are 1,2,3, and 4 therefore product will be 24.

n = 24 + 1 = 25 (We proved that** n is** ** not prime**)

For 'C' we cau put values and find or we can use algebra

(It's easy to find through values)

Using algebra

Let x = y(y + 1)(y + 2)(y + 3)

x = y^{4} + 6y^{3} + 11y^{2} + 6y

Adding and subtracting 1

x = y^{4} + 6y^{3} + 11y^{2} + 6y + 1 - 1

x = (y^{2} + 3y + 1)^{2} - 1

n = x + 1

n = (y^{2} + 3y + 1)^{2} - 1 + 1

n = (y^{2} + 3y + 1)^{2}

**n is perfect square**

**Therefore option 1. A and C only**