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Q.)If n = 1+X where X is the product of four consecutive positive integers , then which of the following is/are true? [CAT Question , CAT 1999 Question , MBA Exam , Number System]

A. n is odd

B. n is prime

C. n is a perfect square

(a) A and C only

(b) A and B only

(c) A only

(d) None of these

Solution : Let the four consecutive integers be (a-1),a,(a+1),(a+2).

X = product of four consecutive positive integers

= (a-1)a(a+1)(a+2)

=$ { (a }^{ 2 }-1)a(a+2)$

= ${ (a }^{ 4 }+2{ a }^{ 3 }-{ a }^{ 2 }-2a)$

n = 1+X = ${ (a }^{ 4 }+2{ a }^{ 3 }-{ a }^{ 2 }-2a)$ + 1

= ${ ({ a }^{ 2 }-2a+1) }^{ 2 }$

X will be even number as it is the product of four consecutive positive integers(one of the integers will be even).

1+X = 1+even = odd

Hence, n is odd and a perfect square also.

Hence, OPTION(a)