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)
Let x = y(y + 1)(y + 2)(y + 3)
x = y4 + 6y3 + 11y2 + 6y
Adding and subtracting 1
x = y4 + 6y3 + 11y2 + 6y + 1 - 1
x = (y2 + 3y + 1)2 - 1
n = x + 1
n = (y2 + 3y + 1)2 - 1 + 1
n = (y2 + 3y + 1)2
n is perfect square
Therefore option 1. A and C only