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CAT Question – Product of Four Consecutive Positive Integers

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)


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